Two-dimensional planar and crossover-free beamforming network architecture

ABSTRACT

An antenna system has a two-dimensional field of view, yet can be implemented on a surface, such as on electronic or photonic integrated circuits. The antenna system includes an array of antennas disposed in a predetermined non-linear pattern and a two-dimensional beamforming network (BFN). The antenna system can be steered/selectively beamformed in two dimensions through beam port selection. The beamforming network is disposed entirely on a single first surface. The beamforming network has a one-dimensional array-side interface disposed on the first surface and a one-dimensional beam-side interface disposed on the first surface. The antennas of the array of antennas are individually communicably coupled to the array-side interface. Segments of the beam-side interface map to respective pixels in the two-dimensional field of view.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication No. 63/068,667, filed Aug. 21, 2020, titled “Two-DimensionalPlanar and Crossover-Free Beamforming Network Architecture,” the entirecontents of which are hereby incorporated by reference herein, for allpurposes.

BACKGROUND Technical Field

The present invention relates to phased array antenna systems and, moreparticularly, to two-dimensional beamforming networks (BFN) for phasedarray antenna systems.

Related Art

Many systems need to emit or receive radio frequency (RF) signals or(humanly visible or invisible) collimated optical beams in directionsthat can be controlled with high precision. For example, such RF oroptical beams are used in radio detection and ranging (RADAR) and lightdetection and ranging (LiDAR) systems, and often these beams need to besteered or swept in one or two dimensions to locate or track a target.Similarly, communications systems, such as fifth generation wirelesscommunications technologies (5G), sometimes need to steer beams, such asto initially establish a line-of-sight communication channel between twoterminals or if one or both of the terminals move.

Phased arrays of RF or optical antenna elements are commonly used insuch systems to form electronically- or optically-steerable directionalsignal beams, i.e., without mechanical steering. One or more receivers,transmitters or transceivers are electrically or optically connected toan array of antenna elements via feedlines, such as optical fibers orcoaxial cables. Taking a transmitter case as an example, thetransmitter(s) operate such that the phase of the signal at each antennaelement is separately controlled. Signals radiated by the variousantenna elements constructively and destructively interfere with eachother in the space (near field) in front of the antenna array. Indirections where the signals constructively interfere, the signals arereinforced, whereas in directions where the signals destructivelyinterfere, the signals are suppressed, thereby creating an effectiveradiation pattern of the entire array in the far field that favors adesired direction. The phases at the various antenna elements, andtherefore the direction in which the antenna array's signal propagates,can be changed very quickly, thereby enabling such a system to besteered, for example to sweep over a range of directions.

According to the reciprocity theorem, a phased array of antenna elementscan also be used to receive signals preferentially from a desireddirection. By dynamically changing the phasing, a system can sweep overa range of directions to ascertain a direction from which a signaloriginates, i.e., a direction from which the signal's strength ismaximum, or to steer a phased array toward a transmitting antenna orother signal source and/or away from interference (noise) sources.

Conventional electronic or optical circuits that control the phases ofsignals fed to, or received from, the antenna elements of a phased arrayare large and complex. A small, thin, and ideally planar, beamformingnetwork (BFN), especially one that fits within a typical mobiletelephone, that can electronically steer or differentiate one or moresignals in two dimensions would be highly desirable. Thus, a problemwith the prior art is how to build a small, thin, and ideally planar,beamforming network that can electronically or optically steer ordifferentiate one or more signals in two dimensions.

SUMMARY OF EMBODIMENTS

In the following summary, parenthetical reference numerals identifynon-limiting exemplary elements described in this Application.

An embodiment of the present invention provides an antenna system. Theantenna system includes an array (402) of antennas and a two-dimensionalbeamforming network (BFN) (420). The array (402) of antennas is disposedin a predetermined non-linear pattern (411). The array of antennasincludes a plurality of antennas (404-410) and has a two-dimensionalfield of view (500). The two-dimensional beamforming network (BFN) (420)is disposed entirely on a single first surface (423). Thetwo-dimensional beamforming network (BFN) (420) has a one-dimensionalarray-side interface (424) disposed on the first surface (423) and aone-dimensional beam-side interface (426) disposed on the first surface(423). The antennas (404-410) of the array (402) of antennas areindividually communicably coupled (412-418) to the array-side interface(424), such that segments (428-432) of the beam-side interface (426) mapto respective pixels (0-15) in the two-dimensional field of view (500).

Optionally, in any embodiment, the array (402) of antennas and thetwo-dimensional beamforming network (420) collectively form a true timedelay system.

Optionally, in any embodiment, the one-dimensional array-side interface(424) is segmented.

Optionally, in any embodiment, the one-dimensional beam-side interface(426) is continuous.

Optionally, in any embodiment, the first surface (423) is planar.

An antenna system according to claim 1, wherein the first surface (423)is non-planar.

Optionally, in any embodiment, the first surface (423) is folded.

Optionally, in any embodiment, the predetermined non-linear pattern(411) defines a second surface (425) that extends smoothly from an edgeof the first surface (423).

Optionally, in any embodiment that defines a second surface, thepredetermined non-linear pattern (411) defines the second surface (425),and the array (402) of antennas (404-410) is communicably coupled to thearray-side interface (424) via a crossover-free network disposedentirely on the first (423) and/or second (425) surface.

Optionally, in any embodiment, the predetermined non-linear pattern(411) defines a second surface (425), and the array (402) of antennas(404-410) comprises a plurality of disjoint sets (600-606) of antennas(404-410). Each disjoint set (600-606) of antennas (404-410) includes aplurality of antennas of the array (402) of antennas (404-410). For eachdisjoint set (600-606) of antennas, each antenna (404-410) of at least anon-empty subset of the disjoint set of antennas is perpendicularlydisplaced (608, 610) a respective distance (614) along the secondsurface (425) from a longitudinal axis (612) of a hypothetical lineararray (700) of antennas disposed on the second surface (425).

Optionally, in any embodiment, the predetermined non-linear pattern(411) defines a second surface (425), and the array (402) of antennas(404-410) includes a plurality of disjoint sets (600-606) of antennas.Each disjoint set (600-606) of antennas includes a plurality of antennasof the array (402) of antennas (404-410). For each disjoint set(600-606) of antennas, each antenna (404-410) of at least a non-emptysubset of the disjoint set (600-606) of antennas is displaced arespective distance (614), measured along the second surface (425) andparallel to one dimension (616, 618) of the two-dimensional field ofview (500), from a longitudinal axis (612) of a hypothetical lineararray (700) of antennas disposed on the second surface (425).

Optionally, in any embodiment in which the array (402) of antennas(404-410) includes a plurality of disjoint sets (600-606) of antennas(404-410), the antenna system has a design wavelength λ. The pluralityof disjoint sets (600-606) of antennas includes N disjoint sets ofantennas. Within each disjoint set (600-606) of the antennas (404-410),the antennas (404-410) are spaced apart in a direction parallel to thelongitudinal axis (612) of the hypothetical linear array (700) ofantennas. Spacing between each pair of adjacent antennas (404-410) is anintegral multiple of about ½ λ. The antennas (404-410) are spaced apartin a direction perpendicular to the longitudinal axis (612) of thehypothetical linear array (700) of antennas.

Optionally, in any embodiment in which the array (402) of antennas(404-410) includes a plurality of disjoint sets (600-606) of antennas(404-410), the antenna system has a design wavelength λ. Within eachdisjoint set (600-606) of the antennas, the antennas (404-410) arespaced apart by respective integral multiples of (λ/2) in the directionparallel to the longitudinal axis (612). The antennas (404-410) arespaced apart by respective integral multiples of (λ/2) in the directionperpendicular to the longitudinal axis (612).

Optionally, in any embodiment, the antenna system has a designwavelength between about 10 nanometers and about 1 millimeter.

Optionally, in any embodiment, the antenna system has a designwavelength between about 1 millimeter and about 100 meters.

Optionally, in any embodiment, the predetermined non-linear pattern(411) defines a second surface (425). Each antenna (404-410) of thearray (402) of antennas includes a grating coupler configured tooptically couple to free space beyond the second surface (425) with acoupling efficiency of at least about 25%.

Optionally, in any embodiment, the two-dimensional beamforming network(420) includes a Rotman lens (422).

Optionally, in any embodiment, the two-dimensional beamforming network(420) includes a Fourier transformer.

Optionally, in any embodiment, the two-dimensional beamforming network(420) includes a Butler matrix.

Optionally, in any embodiment, the two-dimensional beamforming network(420) includes a single-stage beamforming network.

Optionally, in any embodiment, the two-dimensional beamforming network(420) includes a single Rotman lens (422).

Optionally, in any embodiment, the array (402) of antennas (404-410)includes N (N>1) disjoint sets (600-606) of antennas. Each disjoint set(600-606) of antennas includes a plurality of antennas (404-410) of thearray (402) of antennas. The two-dimensional beamforming network (420)includes N first beamforming networks (2302, 2308-2312). Each firstbeamforming network is associated with a distinct set (600-606) of theantennas and has a beam-side interface (2322-2326) and a plurality ofarray-side ports (2328-2334). The array-side ports (2328-2334) of eachfirst beamforming network (2302, 2308-2312) are individuallycommunicably coupled to respective antennas (404-410) of the associatedset of the antennas. The array-side ports (2328-2334) of the N firstbeamforming networks (2302, 2308-2312) thereby collectively form theone-dimensional array-side interface (424) of the two-dimensionalbeamforming network (420). The two-dimensional beamforming network (420)also includes N second beamforming networks (2304, 2314-2318). Eachsecond beamforming network is associated with a distinct firstbeamforming network (2302, 2308-2312) and has an array-side interface(2346-2350) and a beam-side interface (2336-2340). The beam-sideinterface (2336-2340) of each second beamforming network is communicablycoupled to the beam-side interface (2322-2326) of the associated firstbeamforming network (2302, 2308-2312). The two-dimensional beamformingnetwork (420) also includes a third beamforming network (2320) having anarray-side interface (2352) and a plurality of beam-side ports(2356-2360). The array-side interface (2346-2350) of each secondbeamforming network (2304, 2314-2318) is communicably coupled to arespective distinct portion of the array-side interface (2352) of thethird beamforming network (2320). The plurality of beam-side ports(2356-2360) of the third beamforming network (2320) collectively formsthe one-dimensional beam-side interface (426) of the two-dimensionalbeamforming network (420).

Optionally, in any embodiment in which the two-dimensional beamformingnetwork (420) includes N first beamforming networks (2302, 2308-2312), Nsecond beamforming networks (2304, 2314-2318), and a third beamformingnetwork (2320), each first beamforming network (2302, 2308-2312)consists essentially of a respective one-dimensional beamforming network(2362-2364), each second beamforming network (2304, 2314-2318) consistsessentially of a respective one-dimensional beamforming network(2366-2368), and the third beamforming network (2320) consistsessentially of a distinct one-dimensional beamforming network (422).

Optionally, in any embodiment in which the two-dimensional beamformingnetwork (420) includes N first beamforming networks (2302, 2308-2312), Nsecond beamforming networks (2304, 2314-2318), and a third beamformingnetwork (2320), each first beamforming network (2302, 2308-2312)consists essentially of a respective Rotman lens (2362-2364), eachsecond beamforming network (2304, 2314-2318) consists essentially of arespective Rotman lens (2366-2368), and the third beamforming network(2320) consists essentially of a distinct Rotman lens (422).

Optionally, in any embodiment in which the two-dimensional beamformingnetwork (420) includes N first beamforming networks (2302, 2308-2312), Nsecond beamforming networks (2304, 2314-2318), and a third beamformingnetwork (2320), for each first beamforming network (2302, 2308-2312),the array-side ports (2328-2334) of the first beamforming network (2302,2308-2312) are transversely ordered in a first order. The antennas(404-410) of the associated set of the antennas are transversely orderedin a second order. The antennas of the associated set of the antennasare individually communicably coupled to the respective array-side ports(2328-2334) such that the first order is opposite the second order.

Optionally, in any embodiment in which the two-dimensional beamformingnetwork (420) includes N first beamforming networks (2302, 2308-2312), Nsecond beamforming networks (2304, 2314-2318), and a third beamformingnetwork (2320), the antenna system has a design wavelength λ. Thebeam-side interface (2336-2340) of each second beamforming network(2304, 2314-2318) is communicably coupled to the beam-side interface(2322-2326) of the associated first beamforming network (2302,2308-2312) by a respective associated first coupling (2342-2344,2502-2508), thereby collectively defining a plurality of firstcouplings. Each non-central first coupling (2342-2344, 2502-2508) isconfigured to delay signals of wavelength λ propagating therethrough bya respective relative delay amount. The delay amount variesmonotonically transversely across the non-central first coupling(2342-2344, 2502-2508).

Optionally, in any embodiment in which each non-central first coupling(2342-2344, 2502-2508) is configured to delay signals of wavelength λpropagating therethrough by a respective relative delay amount, eachfirst beamforming network (2302, 2308-2312) is numbered with a uniqueinteger j between −[N/2] and +[N/2]. For each non-central first coupling(2342-2344, 2502-2508), the delay amount varies monotonicallytransversely across the non-central first coupling between about+((N−1)/2)jλ and about −((N−1)/2)jλ.

Optionally, in any embodiment in which the delay amount variesmonotonically transversely across the non-central first coupling betweenabout +((N−1)/2)jλ and about −((N−1)/2)j λ, for each central firstcoupling, the relative delay amount is about zero.

Optionally, in any embodiment in which the delay amount variesmonotonically transversely across the non-central first coupling betweenabout +((N−1)/2)jλ and about −((N−1)/2)jλ, each first coupling(2342-2344, 2502-2508) includes a respective plurality of discretewaveguides.

Optionally, in any embodiment in which the delay amount variesmonotonically transversely across the non-central first coupling betweenabout +((N−1)/2)jλ and about −((N−1)/2)jλ, for each first beamformingnetwork (2302, 2308-2312), the set of the antennas associated with thefirst beamforming network (2302, 2308-2312) includes a respective numberM of antennas. The associated first coupling includes M discretewaveguides.

Optionally, in any embodiment in which the delay amount variesmonotonically transversely across the non-central first coupling betweenabout +((N−1)/2)jλ and about −((N−1)/2)jλ, each non-central firstcoupling (2342-2344, 2502-2508) includes a respective medium (4400)configured to delay signals propagating therethrough by a respectivedelay amount. The delay amount varies continuously transversely acrossthe medium.

Optionally, in any embodiment, each dimension of a grating lobe-freetwo-dimensional field of view (502-508) includes a respective number ofpixels. The two-dimensional beamforming network (420) includes aplurality of distinct elementary components. The two-dimensionalbeamforming network (420) is configured such that photons traverse, onaverage, a number of the distinct elementary components that isconstant, with respect to the number of pixels along each dimension ofthe grating lobe-free field of view (502-508).

Another embodiment of the present invention provides an antenna system,the antenna system includes N second Fourier transform-based arrayphasers, and N third Fourier transform-based array phasers. Each secondFourier transform-based array phaser is associated with a distinctdisjoint set of antennas and has a respective array-side interface and arespective beam-side interface. The antennas of each disjoint set ofantennas are communicably coupled by respective non-crossing secondcouplings to the array-side interface of the associated second Fouriertransform-based array phaser. Each third Fourier transform-based arrayphaser is associated with a distinct second Fourier transform-basedarray phaser and has a respective array-side interface and a respectivebeam-side interface. The beam-side interface of each third Fouriertransform-based array phaser is communicably coupled to the beam-sideinterface of the corresponding second Fourier transform-based arrayphaser by a respective non-crossing third coupling. The array-sideinterface of each third Fourier transform-based array phaser iscommunicably coupled to the array-side interface of the first Fouriertransform-based array phaser by a respective non-crossing fourthcoupling. The non-crossing first coupling includes the non-crossingsecond couplings, the N second Fourier transform-based array phasers,the non-crossing third couplings, the N third Fourier transform-basedarray phasers, and the non-crossing fourth couplings.

Optionally, in any embodiment that includes N second Fouriertransform-based array phasers and N third Fourier transform-based arrayphasers, each third coupling includes a respective distinct orderedplurality of third waveguides. Waveguides of each ordered plurality ofthird waveguides are configured to delay signals propagatingtherethrough by respective delay amounts. Each ordered plurality ofthird waveguides is configured such that the delay amount variesmonotonically transversely across the ordered plurality of thirdwaveguides.

Optionally, in any embodiment that includes N second Fouriertransform-based array phasers and N third Fourier transform-based arrayphasers, each fourth coupling includes a respective distinct orderedplurality of fourth waveguides. Waveguides of each ordered plurality offourth waveguides are configured to delay signals propagatingtherethrough by respective delay amounts. Each ordered plurality offourth waveguides is configured such that the delay amount variesmonotonically transversely across the ordered plurality of fourthwaveguides.

Yet another embodiment of the present invention provides an antennasystem. The antenna system includes a first 2-D beamforming network(BFN) and an array of antennas. The first 2-D BFN is disposed on asurface and has an array-side interface and a beam-side interface. Thearray of antennas is disposed in a predetermined non-linear pattern onthe surface and has a two-dimensional field of view. Antennas of thearray of antennas are individually communicably coupled to thearray-side interface of the first BFN coupling disposed on the surface.Segments of the beam-side interface map to respective pixels in thetwo-dimensional field of view.

Optionally, in any embodiment with a non-crossing first coupling, thenon-crossing first coupling includes a plurality of first waveguides.

Optionally, in any embodiment with a first waveguide, each firstwaveguide is configured to provide an equal path length.

Yet another embodiment of the present invention provides an antennasystem. The antenna system includes an array of antennas and N firstRotman lenses. The array of antennas is disposed in a predeterminednon-linear pattern and has a field of view. The array includes N sets(N>1) of the antennas. Each set includes a plurality of the antennas.Each first Rotman lens corresponds to a distinct set of the N sets ofthe antennas. Each antenna is communicably coupled to a respective arrayport of its corresponding first Rotman lens. Respective array ports ofeach first Rotman lens are individually communicably coupled to theantennas of a distinct set of the N sets of antennas by respective firstwaveguides, one first waveguide per antenna. A respective beam side ofeach second Rotman lens is communicably coupled to a respective beamside of a distinct first Rotman lens by a first delay medium having adelay that varies laterally across the medium. An array side of thethird Rotman lens is communicably coupled to respective array sides ofthe second Rotman lenses by respective second delay media. Beam ports ofthe third Rotman lens collectively represent a linear array of pixelsthat maps to an unwound raster scan of pixels of the two-dimensionalfield of view.

Optionally, in any embodiment with a planar array of radiating elementsand a second Rotman lens, the planar array of radiating elements has adesign wavelength. For each second Rotman lens, a difference in delaybetween two maximally different delay line second waveguides is about

$2 \times \left( \frac{N - 1}{2} \right)^{2} \times {the}{design}{}{{wavelength}.}$

Optionally, in any embodiment with delay line second waveguides and asecond Rotman lens, for each second Rotman lens, all pairs of adjacentdelay line second waveguides have equal (within manufacturingtolerances) differences in delay.

Optionally, in any embodiment with radiating elements, the radiatingelements are equally spaced apart (within manufacturing tolerances)along each row and along each column, and intra-element row spacing ofthe radiating elements is about equal to intra-element column spacing ofthe radiating elements.

Optionally, in any embodiment with a planar array of radiating elements,rows and columns of the planar array of radiating elements is tilted,relative to a raster scan of the pixels of the two-dimensional field ofview, at a slope of about

$\frac{1}{N}.$

Optionally, in any embodiment with a planar array of radiating elements,the planar array of radiating elements has a design wavelength betweenabout 100 meters and about 1 millimeter.

Optionally, in any embodiment with a planar array of radiating elements,the planar array of radiating elements has a design wavelength betweenabout 10 nanometers and about 1 millimeter.

Optionally, in any embodiment with a Rotman lens, for each second Rotmanlens, N delay line second waveguides are coupled to the second Rotmanlens and arranged in an S-bend design. The N delay line secondwaveguides are parallel (within manufacturing tolerances) to each other.Each delay line second waveguide includes a respective aboutthree-quarter-circle curve. The N delay line second waveguides therebycollectively form a plurality of concentric, equally radiallyspaced-apart about three-quarter-circle curves. Each delay line secondwaveguide includes a respective about half-circle curve that extends ina direction opposite the about three-quarter-circle curve of the delayline second waveguide and has a radius different from the radius of theabout half-circle curve of each other delay line second waveguide. The Ndelay line second waveguides thereby collectively form a plurality ofabout half-circle curves.

Optionally, in any embodiment with a Rotman lens, for each second Rotmanlens, half-circle curves of a plurality of about half-circle curves arenon-concentric, and centers of the half-circle curves are co-linear.

Optionally, in any embodiment with a second Rotman lens, for each secondRotman lens, the half-circle curves of the plurality of abouthalf-circle curves are concentric.

Optionally, in any embodiment with a Rotman lens, delay line secondwaveguides are coupled to a second Rotman lens are arranged in an S-benddesign. The path of each delay line second waveguide progresses into ahalf circle shape and a sloped half circle shape. The half circle andthe sloped half circle shapes have a different radius for each delayline second waveguide. Longer delay line second waveguides fit aroundshorter delay line second waveguides.

Optionally, in any embodiment with delay line second waveguides, thehalf circle and sloped half circle shapes of left delay line secondwaveguides coupled to a second Rotman lens face opposite directions fromright delay line second waveguides coupled to the second Rotman lens.

Optionally, in any embodiment with delay lines, path lengths of longerdelay line second waveguides are accommodated by extending bends of thehalf circle shape of each delay line second waveguide, and theextensions of the bends become increasingly pronounced as progressingfrom the center delay line second waveguide toward the left or rightdelay line second waveguides (“tromboning”).

Optionally, in any embodiment with delay line waveguides, the N delayline second waveguides coupled to a second Rotman lens have a width ofabout (3N-1) times spacing between a pair of N delay line secondwaveguides, plus a diameter of the half circle shape with minimalradius.

Optionally, in any embodiment with a Rotman lens, a waveguide enters andexits each Rotman lens orthogonal to the surface of the Rotman lens.

Optionally, in any embodiment, the array of antennas is planar.

Optionally, in any embodiment, the array of antennas is organized as Nrows and N columns.

Optionally, in any embodiment, first waveguides have equal path lengths.

Optionally, any embodiment includes respective sets of N progressivelylonger delay line second waveguides.

Optionally, any embodiment includes a plurality of equal-path-lengththird waveguides.

Optionally, any embodiment includes N third waveguides per second Rotmanlens.

Optionally, in any embodiment, each antenna is communicably coupled tothe respective array port of its corresponding first Rotman lens via arespective first waveguide.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fee.

The invention will be more fully understood by referring to thefollowing Detailed Description of Specific Embodiments in conjunctionwith the Drawings, of which:

FIG. 1 is a diagram of a phased array of antennas, according to theprior art.

FIG. 2 is a diagram of a Rotman lens used as a phaser for a linear(one-dimensional) array of antennas, according to the prior art.

FIG. 3 is a diagram of a two-stage stacked Rotman lens-basedtwo-dimensional beamformer, according to the prior art.

FIG. 4 is a partially schematic, top view diagram of a crossover-freeantenna system capable of two-dimensional beamforming, according to anembodiment of the present invention.

FIG. 5 is a diagram of an exemplary mapping of far field pixel-to-portnumber of a beam-side interface of a beamformer in the antenna system ofFIG. 4 , according to an embodiment of the present invention.

FIG. 6 is an enlarged view of portion, specifically an antenna array, ofthe antenna system of FIG. 4 , according to an embodiment of the presentinvention.

FIG. 7 is a diagram of a hypothetical linear array of antennas disposedon a surface, according to the prior art.

FIG. 8 illustrates a conventional linear phased array of antenna, suchas the phased array of FIG. 1, 2 or 7 , and its ability to form beamsthat are limited to bands in the antenna's field of view, according tothe prior art.

FIGS. 9-15 illustrate how displacing the antennas of the antenna systemof FIG. 4 from a longitudinal axis (for example, the linear array ofFIG. 7 ) affects the mapping of the ports of the beam-side interface torespective portions (pixels) of the (two-dimensional) field of view,according to an embodiment of the present invention.

FIG. 16 illustrates how changing tilt of the antennas of the antennasystem of FIGS. 4 and 9-15 impacts tilt of the pixels of the field ofview, according to an embodiment of the present invention.

FIG. 17 illustrates a path meandering arrangement for a compactpath-matching network, according to an embodiment of the presentinvention.

FIG. 18 illustrates how each feedline of FIG. 17 can be lengthened by anidentical amount, without increasing width of the meanderingarrangement, according to an embodiment of the present invention.

FIG. 19 is similar to FIG. 18 , except each feedline is lengthened by adifferent amount, according to an embodiment of the present invention.

FIG. 20 illustrates a path meandering arrangement, similar to the pathmeandering arrangement of FIGS. 17-19 , except mirrored left-to-right,according to an embodiment of the present invention.

FIG. 21 illustrates dimensions, coordinates and mathematicalrelationships there among for designing the compact path-matchingnetwork of FIGS. 17-20 , according to an embodiment of the presentinvention.

FIG. 22 shows an enlarged portion of FIG. 21 .

FIG. 23 is a partially schematic diagram of a crossover-free antennasystem capable of two-dimensional beamforming, according to anembodiment of the present invention.

FIG. 24 illustrates a column-wise longitudinal reversal of antenna orderand resulting change in pixel arrangement of the crossover-free antennasystem of FIGS. 4, 6 and 23 , according to an embodiment of the presentinvention.

FIG. 25 is a partially schematic diagram of a crossover-free antennasystem capable of two-dimensional beamforming, according to anembodiment of the present invention.

FIG. 26 illustrates imposing delays between stages of the crossover-freeantenna system and resulting effect on broadband pixels, according to anembodiment of the present invention.

FIG. 27 illustrates an exemplary meandering arrangement of feedlines forimplementing the delays of FIG. 26 , according to an embodiment of thepresent invention.

FIGS. 28-34 illustrate how progressively increasing relative delays inthe feedlines progressively shifts the pixels from a one-dimensionalbeamforming arrangement to a two-dimensional beamforming arrangement,according to an embodiment of the present invention operating inbroadband.

FIG. 35 illustrates a tilting of antennas and resulting tilt in pixelsof the crossover-free antenna system of FIGS. 4, 6 and 23 , according toan embodiment of the present invention.

FIGS. 36-37 illustrate a similar relationship between tilting antennasand tilt in pixels, for a larger array of antennas than in FIG. 35 ,according to an embodiment of the present invention.

FIG. 38 illustrates an inverse relationship between antenna spacing andpixel size, according to an embodiment of the present invention.

FIGS. 39-41 illustrate a similar relationship between antenna spacingand pixel size, for a larger array of antennas than in FIG. 38 ,according to an embodiment of the present invention.

FIG. 42 is a scale drawing of a prototype 4×4 crossover-free antennasystem capable of two-dimensional beamforming, fabricated on a photonicintegrated circuit, according to an embodiment of the present invention.

FIG. 43 contains a table of relative path delays between first andsecond stage beamforming networks for an exemplary 7×7 antenna array,according to an embodiment of the present invention.

FIG. 44 illustrates an electromagnetic coupling between first and secondbeamforming networks provided by a continuous bulk medium, rather thanindividual waveguides, according to an alternative embodiment of thepresent invention.

FIG. 45 illustrates two variations of the meandering arrangement of FIG.17 , according to respective embodiments of the present invention.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

Embodiments of the present invention provide antenna systems that havetwo-dimensional fields of view, yet can be implemented on surfaces, suchas on electronic or photonic integrated circuits. Each such antennasystem includes an array of antennas disposed in a predeterminednon-linear pattern and a two-dimensional beamforming network (BFN).“Two-dimensional” in this context means the array of antennas has atwo-dimensional field of view and can differentiate points within thefield of view with two degrees of freedom. The antenna system can besteered in two dimensions through beam port selection. The beamformingnetwork is disposed entirely on a single first surface. The beamformingnetwork has a one-dimensional array-side interface disposed on the firstsurface and a one-dimensional beam-side interface disposed on the firstsurface. The antennas of the array of antennas are individuallycommunicably coupled to the array-side interface. Segments of thebeam-side interface map to respective pixels in the two-dimensionalfield of view. In some embodiments, no signal crosses another signal inthe BFN.

The antenna system may be used in mobile telephones, automotive LiDARand other systems, particularly compact systems, in which directionaltransmission and/or reception of electromagnetic (EM) waves isdesirable. For example, imaging systems (“cameras”) can be thought of ascollecting light from a discrete set of directions (“pixels”)simultaneously or nearly simultaneously. Thus, the antenna system may beused in cameras. Signals to and/or from the beam-side interface may beprocessed by circuits that are also disposed on the first surface, orthese signals may be carried by suitable waveguides to an edge of thefirst surface and there coupled to external circuits.

Definitions

A “beamforming network” is a network of waveguides that connects anarray of C couplers to P output ports, such that a plane wave (as thatterm is commonly used by one of ordinary skill in the art) incident onthe couplers from a particular direction gets substantially concentratedinto a single output port. According to the reciprocity theorem,radiation input into a port is distributed among the couplers with suchamplitudes and phase relationships that emissions from the couplers forma plane wave in a particular direction (a beam). A “coupler” is a devicethat interfaces waves propagating through a medium (including freespace) and a signal carried by a lead, such as a waveguide. Examplesinclude: metal conductors that interface between EM waves propagatingthrough space and electric currents moving in the metal conductors;rectennas, including optical rectennas; optical gratings or mirrorsdisposed between waveguides and free space; sonic or ultrasonictransducers; and the like. For generality, couplers are referred toherein as antennas.

As used herein, EM waves include, but are not limited to, radiofrequency (RF) signals (roughly about 20 kHz to about 300 GHz or about 1mm to about 15 km in wavelength) and humanly visible and invisibleoptical signals (about 10 nm to about 1 mm in wavelength). Antennasystems are described herein in the context of EM waves. However,principles described herein apply to wavelengths longer or shorter thanEM wavelengths, as well as to other types of waves, such as acousticwaves. Use of these principles at these other wavelengths or with theseother types of waves is explicitly contemplated.

As described in Wikipedia, an antenna array (or array antenna or phasedarray) is a set of multiple connected antennas that work together as asingle antenna, to transmit or receive EM waves. FIG. 1 is a diagram ofa phased array of antennas A, according to the prior art. The individualantennas (elements A) may each be connected to (fed or driven by) arespective receiver or transmitter, or multiple individual elements Amay be fed by a single receiver or transmitter (TX in FIG. 1 ). Theelements A are fed in a specific static or dynamic phase relationship.The connections may include respective feedlines, represented byfeedline 100, that feed to, or receive from, the elements A. “Driven”means connected to the receiver(s) and/or transmitter(s), as opposed toparasitic. An antenna array may, but need not, include parasiticelements, such as reflectors and/or directors, in addition to drivenelements.

“Design wavelength” means a wavelength at which a device is designed tooperate, or a center of a range of wavelengths over which the device isdesigned to operate. The range of wavelengths, over which the device isdesigned to operate, is referred to as a “design bandwidth.”

In FIG. 1 , the phase of the signal to each element A is controlled by arespective delay circuit (Φ in FIG. 1 ). The delay circuits Φ arecontrolled by a phase controller C. As a result of the specific feedphase relationship, the EM waves radiated by the respective individualantennas A combine and superpose, adding together (interferingconstructively) to enhance the power radiated in desired directions,exemplified by arrow 102, and cancelling (interfering destructively) toreduce the power radiated in other directions. Although a singledirection 102 is indicated in FIG. 1 , some antenna arrays have multiplelobes, each aimed in a different direction. When used for receiving,separate EM signals from the individual antennas A combine with thecorrect phase relationship to enhance signals received from the desireddirections and cancel signals from undesired directions.

Problematically, applying the same delay Φ to all frequencies on a givenfeedline 100 creates a phenomenon known as “beam squint,” which is afrequency-dependent distortion of the beam steering angle. The phaseshift at the low end of a signal's spectrum and the phase shift at thehigh end of the signal spectrum are different enough that the beampoints differently from one extreme to the other. Narrowband signals donot experience beam squint, but application often require broadbandoperation.

Various techniques are known for controlling the phase to each elementA. In general, these techniques are referred to as involving beamformingnetworks (BFN). One such technique, illustrated in FIG. 2 , involvescoupling a Rotman lens 200 (sometimes called the Rotman-Turner lens)between a transmitter or receiver and a linear array of antennaelements, exemplified by antenna elements 202, 204, 206 and 208. ARotman lens 200 is a type of beam-forming network that allows multiplebeams to be formed without the need for switches or phase shifters. TheRotman lens 200 uses RF guided waves in a specially designed geometricalstructure to produce delays for a number of discrete beams. The antennaelements 202, 204, 206 and 208 are connected to the right side (as shownin FIG. 2 ) array ports, with beam ports connected to the left.

Parallel plates (parallel to the plane of FIG. 2 ) define a cavitytherebetween, through which signals propagate between the beam ports andthe array (antenna) ports. Signals radiated within the Rotman lens by agiven beam port, for example beam port 4, have different path lengths tothe various antenna ports. Thus, the signals received at the variousantenna ports, from the beam ports, have various phases, relative toeach other. The beam and antenna ports are disposed, relative to eachother, such that the phase shifts experienced by the antenna ports areappropriate for feeding the antenna elements.

Each beam port causes the antenna array to be phased (directed) in adifferent direction. Thus, the Rotman lens might be thought of as aquasi-microstrip or a quasi-stripline circuit, where the beam ports arepositioned such that constant phase shifts are achieved at the antennaports. Consequently, a Rotman lens concentrates EM signals received atthe array ports with a particular phase slope (equivalently, a planewave incident with a particular angle of arrival), into a particularbeam port. We refer to this property of the Rotman lens as “analyzingthe angle of arrival” of the incident wave.

Although a Rotman lens may have many beam ports, the beam ports areisolated from each other, in that the beam ports do not greatly affectthe loss, or noise figure, of adjacent beams. A well-designed Rotmanlens may have as little as 1 dB of loss. However, the Rotman lenscircuit of FIG. 2 is limited to beamforming in a single dimension, i.e.,along the X axis.

Multiple Rotman lenses may be combined, as shown in the left portion ofFIG. 3 , to form a two-stage stacked two-dimensional (X-Y) beamformingnetwork that feeds a uniform rectangular array (URA) of antennas. Thecenter portion of FIG. 3 (“FIG. 3 Continued”) shows radiation patternsthat can be generated by the beamformer. The right portion of FIG. 3(“FIG. 3 Continued”) provides a comparison of the two-stage stackedarrangement with a single-stage Rotman lens-based beamformer.

However, each stage requires a three-dimensional (X-Y-Z) structure,which would be difficult or impossible to house in a 5G mobile telephoneor automotive LiDAR unit. Although, arguendo, it may be possible toreconfigure the two-stage stacked beamformer of FIG. 3 on a singlesurface, such a reconfiguration would require signal lines to cross overeach other, which is difficult to fabricate and would most likely causeunacceptable losses and/or noise.

A Butler matrix is another well-known device for beamforming. A Butlermatrix, first described by Jesse Butler and Ralph Lowe in “Beam-FormingMatrix Simplifies Design of Electronically Scanned Antennas,” ElectronicDesign, volume 9, pp. 170-173, Apr. 12, 1961, the entire contents ofwhich are hereby incorporated by reference herein for all purposes, is atype of passive phasing network having N inputs and N outputs, usually apower of two. A Butler matrix, coupled between a set of antenna elementsand a transmitter or a receiver, may be used for beamforming.

The N inputs of a Butler matrix are isolated from each other. Phases ofthe N outputs are linear, with respect to position, so the beams areevenly spaced. The phase increments, among the outputs, depend on whichinput is used. For example, a Butler matrix may be constructed such thatwhen input port 1 is used, the four outputs are linearly phased in 45degree increments; when input port 2 is used, the four outputs arelinearly phased in 135 degree increments; when input port 3 is used, thefour outputs are linearly phased in 225 degree increments; and wheninput port 4 is used, the four outputs are linearly phased in 315 degreeincrements. Thus, depending on which of the N inputs is accessed, theantenna beam is steered in a specific direction in one plane. Thus, thebeam steering is in one dimension. Two Butler matrices can be combinedto facilitate two-dimensional steering. However, such a combination,like the two-stage stacked Rotman lens configuration of FIG. 3, is toobulky for use in a 5G mobile telephone or automotive LiDAR unit.

A significant feature of a Rotman lens is that it operates by adjustingtotal distance traveled by EM radiation inside the device, such that allpaths taken by the EM radiation from (to) a particular direction (port)are equal in effective length. This property allows the device tooperate over a wide range of wavelengths (has broad bandwidth) and is,therefore, referred to as a “true time delay” (TTD) device. Unlike aRotman lens, a Butler matrix only equalizes phase, i.e., path length,modulo the wavelength. Butler matrices are, therefore, intrinsicallynarrowband devices.’

A device is said to have true time delay if the time delay through thedevice is independent of the frequency of the EM signal. TTD is animportant characteristic of lossless and low-loss, dispersion free,transmission lines. TTD allows for a wide instantaneous signal bandwidthwith virtually no signal distortion, such as pulse broadening, duringpulsed operation.

Two-Dimensional Planar and Crossover-Free Beamforming NetworkArchitecture

FIG. 4 is a diagram of a crossover-free antenna system 400 capable oftwo-dimensional beamforming, according to an embodiment of the presentinvention. The antenna system 400 includes an array 402 of antennas(outlined in dashed line). However, unlike the prior art, the antennasystem 400 can be implemented on a single surface 403. “Two-dimensional”in this context means the array of antennas 402 has a two-dimensionalfield of view and can differentiate points within the field of view withtwo degrees of freedom, for example elevation and azimuth. A diagram ofthe field of view 500 is described herein, with reference to FIG. 5 . Inthe embodiment shown in FIG. 4 , the two-dimensional field of view isout of the plane of the diagram, i.e., generally toward the reader. Asused herein, “surface” means a piecewise smooth surface, i.e., acontinuous surface that may include an assembly of smaller smoothsurfaces, where interfaces between adjacent smaller smooth surfaces mayform sharp edges. A three dimensional component, such as a lens, that ismounted on a surface such that only a thin or planar or nearly planarslice of the component is used for purposes described hereinnevertheless meets the requirements of being disposed on the surface,since portions of the component above and/or below the surface aresuperfluous, at least with respect to the beamforming network describedherein. A surface can, but need not necessarily, be planar.“Crossover-free” means no feedline crosses another feedline.Cross-coupling within a Rotman lens or other Fourier transformer is notconsidered a crossover, within the context of the present application.

The array of antennas 402 includes a plurality of individual antennas,exemplified by antennas 404, 406, 408 and 410. Although the embodimentshown in FIG. 4 includes 16 antennas 404-410, other embodiments caninclude fewer or more antennas. Importantly, the antennas 404-410 aredisposed in a predetermined non-linear pattern. For example, in theembodiment shown in FIG. 4 , the antennas 404-410 are disposed in aparallelogram array, as indicated by dotted line 411. Antenna 404-410arrangement is discussed in more detail herein. In some embodiments, theantennas 404-410 are disposed in a planar array. A planar array isreferred to herein as being disposed on a planar surface, although thesurface may be merely conceptual, not necessarily physical. In someembodiments, the antennas 404-410 are disposed in a curved array. Acurved array is referred to herein as being disposed on a curvedsurface, such as a portion of a sphere, although the surface may bemerely conceptual, not necessarily physical.

If the antenna system 400 is to be used at radio frequencies, theantennas 404-410 may, for example, be electrically conductive dipole orpatch antenna elements of appropriate lengths/sizes, based on a designwavelength λ of the antenna system 400. On the other hand, if theantenna system 400 is to be used at optical wavelengths, the antennas404-410 may, for example, be mirrors or optical gratings havingappropriate groove spacings, based on the design wavelength λ of theantenna system 400, to optically couple a feedline to a medium, such asair, oil or vacuum beyond the mirror or optical grating (for simplicityof explanation, collectively referred to herein as “free space”), with acoupling efficiency of at least about 25%. In embodiments where opticalwaveguides extend perpendicular to a desired direction of free spacepropagation, an optical coupler should facilitate this change ofdirection. Examples of optical couplers include compact gratings, prismsfabricated in waveguides and facets etched in wafers and used asmirrors. Optical antennas are described by Palash Bharadwaj, et al.,“Optical Antennas,” Advances in Optics and Photonics 1.3 (2009), pp.438-483, U.S. Pat. Publ. 2014/0192394 to Jie Sun, et al., “OpticalPhased Arrays” (2014), and U.S. Pat. Publ. No. 2018/0175961 to Steven J.Spector, et al., “Integrated MEMS Switches for Selectively CouplingLight In and Out of a Waveguide” (2018), the entire contents of each ofwhich are hereby incorporated by reference herein, for all purposes. Insome embodiments, the antennas 404-410 are fabricated on an electronicor photonic integrated circuit.

An antenna or antenna element is fed by a “feedline,” also referred toherein as a transmission line. Examples of feedlines include coaxialcables, striplines, hollow microwave waveguides and other waveguides,such as sufficiently optically transparent materials bounded by othermaterials having suitable indexes of refraction to contain EM waves inthe optically transparent materials by means of total internalreflection, for example optical fibers and optical waveguides fabricatedon optical integrated circuits. Of course, suitability of a feedlinedepends on wavelength and power of a signal to be carried by thefeedline.

Each antenna 404-410 is coupled to a respective feedline, exemplified byfeedlines 412, 414, 416 and 418, appropriate for the design wavelengthk, power, etc. Suitable feedlines 412-418 include coaxial cables, striplines, optical fibers, sufficiently optically transparent materialsbounded by other suitable materials, or other waveguides based, ofcourse, on the design wavelength λ and power. In some embodiments, thefeedlines 412-418 are fabricated on an electronic or photonic integratedcircuit. Additional information about possible feedlines, including pathlengths of the respective feedlines 412-418, is provided herein. In anycase, the antennas 404-410 are suitable for emitting electromagneticwaves, at the design wavelength k, from their respective feedlines412-418 into the medium, such as free space, and/or receivingelectromagnetic waves from the medium into their respective feedlines412-418.

The antenna system 400 also includes a two-dimensional beamformingnetwork 420. “Two-dimensional” in this context means the beamformingnetwork 420 enables the array of antennas 402 to form a beam directedinto the two-dimensional field of view, with two degrees of freedom ofdirectionality of the beam, for example horizontal and vertical, asindicated by respective arrows in FIG. 4 . In the embodiment shown inFIG. 4 , the beamforming network 420 includes a single Rotman lens 422.In other embodiments, the beamforming network 420 includes a suitableButler matrix (not shown) or any suitable Fourier transformer (notshown). In yet other embodiments, the beamforming network 420 includesadditional components, as described herein.

In some embodiments, the feedlines 412-418 provide equal (withinmanufacturing tolerances) effective path lengths between the respectiveantennas 404-410 and the beamforming network 420 and/or the Rotman lens422. In some embodiments, the antenna system 400 is a true time delaydevice. “Effective path length” (in optics, commonly referred to asoptical path length (OPL)) takes into consideration physical length of apath and velocity factor (index) of media in the feedlines 412-418,through which the EM signals propagate. “Equal effective path length”means EM signals at the design wavelength λ, or within the designbandwidth, take essentially equal amounts of time to propagate throughthe respective feedlines 412-418. Differences in the effective pathlengths are sufficiently small that signals received or transmitted byrespective single antennas 404-410 do not involve phase differences inthe antennas' feedlines 412-418 that would cause beam forming directionerrors greater than a design angular resolution of the antenna system400. In other words, for an antenna system 400 designed to resolve X andY direction of received and/or transmitted signals with a given angularresolution, the effective path lengths of the feedlines 412-418 aresufficiently close to each other not to introduce phase differences thatwould cause beam forming direction errors greater than the designangular resolution of the antenna system 400. For simplicity, FIG. 4does not show these equal effective path lengths. Embodiments of thefeedlines 412-418 are described in more detail herein.

In some embodiments, the beamforming network 420 is disposed entirely ona single first surface 423. In some embodiments, the beamforming network420 is fabricated on an electronic or photonic integrated circuit. Insome embodiments, the predetermined non-linear pattern of the antennas404-410 defines a second surface 425 that extends continuously from anedge of the first surface 423. A boundary 427 between the first 423 andsecond 425 surfaces may be arbitrary. For example, the first and secondsurfaces 423 and 425 may be respective portions of a larger electronicor photonic integrated circuit or other planar or non-planar surface. Insome embodiments, the beamforming network 420 is planar, i.e., the firstsurface 423 is planar. In some embodiments, the array of antennas 402 isco-planar with the beamforming network 420. A planar beamforming network420 is referred to herein as being disposed on a planar surface,although the surface may be merely conceptual, not necessarily physical.In some embodiments, the beamforming network 420 is disposed on anon-planar surface, for example a curved or folded surface, although thesurface may be merely conceptual, not necessarily physical. In any case,the beamforming network 420 is thin, preferably no thicker than anelectronic or photonic integrated circuit. In some embodiments, thethickness of the beamforming network 420 may be even thinner than thefree space design wavelength λ of the antenna system 400. Designwavelength is typically defined as free space wavelength. By using highindex materials, signals may be more tightly confined in the materialsthan in free space. Thus, in some embodiments, the electronic orphotonic integrated circuit may be thinner than the free space designwavelength.

Several embodiments of the beamforming network 420 are described herein.However, in any embodiment, the beamforming network 420 has aone-dimensional array-side interface 424 disposed on the first surface423 and a one-dimensional beam-side interface 426 disposed on the firstsurface 423. In some embodiments, such as the embodiment shown in FIG. 4, the one-dimensional beam-side interface 426 is segmented. “Segmented”in this context means the beam-side interface has individual ports,exemplified by ports 428, 430 and 432. However, in some embodiments, theone-dimensional beam-side interface 426 is continuous, i.e., lackingdiscrete ports. Nevertheless, portions of a continuous beam-sideinterface 426 may be referred to as segments of the beam-side interface426. For example, a one-dimensional continuous beam-side interface 426may be fed by a one-dimensional row of lasers, such as one laser persegment, or one or more lasers able to be moved or swept across thecontinuous interface, or a continuous beam-side interface 426 may feed aone-dimensional row of optical or RF sensors or a single movable sensor.A “one-dimensional” interface means a single, possibly curved, i.e., notnecessarily linear, row interface, not multiple stacked rows, such as afour row by 16 port interface.

The antennas 404-410 of the array of antennas 402 are individuallycommunicably coupled, via the feedlines 412-418, to the array-sideinterface 424. “Individually communicably coupled” means each antenna404-410 is coupled by a distinct feedline 412-418 to the array-sideinterface 424. No antenna 404-410 couples to more than one feedline412-418. Thus, for example, a series-fed microstrip patch antenna arraythat is steered by varying wavelength of a signal fed to the array via asingle feedline is explicitly not an array of antennas, in which theantennas are individually communicably coupled. In the embodiment shownin FIG. 4 , no single feedline 412-418 couples more than one of theantennas 404-410 to the array-side interface 424.

Each antenna 404-410 may, however, include multiple elements or multipleantennas (for clarity, referred to herein as elements), such as toincrease signal capture/emission area of the antenna over that of asingle element. Each element is configured to emit and/or receive EMwaves. For example, individual antennas 404-410 may be made up ofrespective H-trees of elements. However, all the elements of a givenantenna 404-410 are spaced sufficiently close together, such that asignal received or transmitted by the elements of a single antenna404-410 does not involve phase differences in that antenna's feedline412-418 that would cause beam forming direction errors greater than adesign angular resolution of the antenna system 400. In other words, foran antenna system 400 designed to resolve X and Y direction of receivedand/or transmitted signals with a given angular resolution, antennaelements are spaced sufficiently close together not to introduce phasedifferences that would cause beam forming errors greater than the designangular resolution of the antenna system 400.

Segments, exemplified by beam ports 428, 430 and 432, of a segmentedbeam-side interface 426 map to respective pixels in the two-dimensionalfield of view. Similarly, respective portions (also referred to as“segments”) of a continuous beam-side interface 426 map to correspondingregions in the two-dimensional field of view. FIG. 5 is a diagram of anexemplary mapping 500 of far field pixel intensity-to-port number428-432 of the segmented beam-side interface 426. The mappingessentially describes a field 500 of view of the antenna system 400. Xand Y axes are indicated in the mapping 500 as sin(X angle) and sin(Yangle). Individual pixels of the field of view are shown as respectiveellipses. Each ellipse is numbered 0, 1, 2, 3, . . . , 15 to correspondto one of the ports 428-432 of the port-side interface 426 (FIG. 4 ).The number of pixels 0, 1, 2, 3, . . . , 15 is equal to the number ofports 428-432. It should be noted that many of the pixels 0, 1, 2, 3, .. . , 15 are duplicated, as grating lobes, when the antennas are spacedapart further than λ/2. Thus, a square area (subset) of the field ofview 500, for example square area 502 shown in heavy dashed line, may becompletely covered (mapped) by the entire set of beam ports 428-432 ofthe beam-side interface 426. Square area 504 is also covered (mapped) bythe entire set of beam ports 428-432. Indeed, some subsets of the fieldof view, for example square area 506, overlap other subsets.Furthermore, a subset of the field of view, for example square area 508,need not necessarily begin with pixel 0 in a corner.

Thus, individual pixels of a two-dimensional field of view 500 may beaccessed, for transmission and/or reception of EM waves, via theone-dimensional beam-side interface 426 (FIG. 4 ). Furthermore, theantenna array 402 and the beamformer 420 can each be disposed on arespective surface 425 and/or 423, or the same surface. Each surface 425and/or 423, or the single surface, can be planar. In any case, theantenna array 402 and the beamformer 420 can be thin, in someembodiments as thin as an electronic or photonic integrated circuit, andin some embodiments thinner than the design free space wavelength λ.

In contrast, the two-stage stacked Rotman lens configuration of FIG. 3requires several Rotman lenses to be stacked in its first stage, andseveral additional Rotman lenses to be stacked in its second stage. Thetwo stages are disposed one behind the other (in the z direction),behind the uniform rectangular array of antennas. Even if, arguendo,each Rotman lens could be as thin as an electronic or photonicintegrated circuit, each stack in FIG. 3 is considerably taller than thebeamforming network 420 (FIG. 4 ) described herein. Thus, the two-stagestacked Rotman lens configuration of FIG. 3 is a three-dimensionalstructure, whereas the beamforming network 420 is a two-dimensionalstructure. In this context, “two-dimensional” means having length andwidth, but essentially no height greater than a single electronic orphotonic circuit, and “three-dimensional” means having length, width andheight at least 1/100 the lesser of the width or the height. Thus, thebeamforming network 420 can be fitted inside a 5G mobile telephone orautomobile LiDAR unit, whereas the two-stage stacked Rotman lensconfiguration of FIG. 3 cannot practically be used in such a mobiledevice.

Furthermore, embodiments of the beamforming network 420 can be madeinexpensively, in large quantities, using conventional electronic orphotonic integrated circuit fabrication techniques, such asphotolithography, on appropriate substrates, such as silicon wafers,whereas the two-stage stacked Rotman lens configuration of FIG. 3 is notsuitable for such fabrication alone. The two-stage stacked Rotman lensconfiguration of FIG. 3 requires forming two stacks, orienting the twostacks perpendicular to each other and interconnecting the two stackstogether and to the uniform rectangular array of antennas. These stepscannot be performed using only conventional electronic or photonicintegrated circuit fabrication techniques.

Antenna Array

As noted, the antennas 404-410 (FIG. 4 ) are disposed in a predeterminednon-linear pattern. A conventional phased array of antennas that uses asingle Rotman lens for phasing includes a linear array of antennas, asshown in FIG. 2 . In contrast, in embodiments of the present invention,the array of antennas 402 includes a plurality of disjoint sets ofantennas. Collectively, the disjoint sets of antennas are disposed in apredetermined non-linear pattern. FIG. 6 is an enlarged view of aportion of FIG. 4 , in particular the array of antennas 402. In theembodiment shown in FIG. 6 , the array of antennas 402 includes fourdisjoint sets of antennas, specifically sets 600, 602, 604 and 606, asindicated by dashed rectangles. “Disjoint set” means no antenna is inmore than one set.

Each set 600-606 includes a plurality of antennas of the array ofantennas 402. For example, set 606 includes four antennas, includingantennas 404 and 406. Although the embodiment shown in FIG. 6 includesfour sets 600-606 and four antennas per set, other embodiments caninclude fewer or more sets and/or fewer or more antennas per set. Insome embodiments, the number of antennas in each set 600-606 equals thenumber of sets 600-606. In other embodiments, the number of antennas ineach set 600-606 is not equal to the number of sets 600-606. In someembodiments, each set 600-606 contains the same number of antennas.However, in other embodiments, each set 600-606 need not necessarilycontain the same number of antennas. Furthermore, some embodimentsinclude sparse arrays of antennas. Although the embodiment shown in FIG.4 has the antennas 404-410 disposed in a parallelogram 411, in otherembodiments, the antennas 404-410 may be disposed in other shapes, suchas squares, rectangles, or rotated versions thereof.

The predetermined non-linear pattern may define the second surface,although the second surface may be merely conceptual, not necessarilyphysical. For each disjoint set 600-606 of antennas, each antenna of atleast a non-empty subset, for example antennas 404 and 406 of set 600,of the disjoint set of antennas may be perpendicularly displaced, asindicated by arrow 608 or arrow 610, a respective distance along thesecond surface from a longitudinal axis 612 of a hypothetical lineararray of antennas (described below) disposed on the second surface. Inother words, at least some of the antennas of each set 600-606 areperpendicularly displaced from the longitudinal axis 612. For example,antenna 408 is perpendicularly displaced a distance 614 from thelongitudinal axis 612.

In another embodiment, for each disjoint set 600-606 of antennas, eachantenna of at least a non-empty subset of the disjoint set of antennasis displaced a respective distance, measured along the second surfaceand parallel to one dimension 616 or 618 of the two-dimensional field ofview, from a longitudinal axis 612 of a hypothetical linear array 700(FIG. 7 ) of antennas disposed on the second surface.

FIG. 7 shows a hypothetical linear array 700 of antennas, exemplified byantennas 702, 704, 706 and 708, disposed on the second surface and alongthe longitudinal axis 612. As shown in FIG. 2 , the phase differencesbetween the antennas 702-708 varies monotonically along the entirelinear array 700 of antennas, and the phase differences depend on anangle θs (FIG. 2 ), at which a wave approaches, or is emitted by, thelinear array of antennas 700.

In contrast, a wave approaching the non-linear pattern of FIG. 6 from adirection other than normal to the second surface causes generation ofsignals in the various feedlines 412-418 whose phases do not varymonotonically along the entire non-linear array of antennas 402. Thestructure of the antenna system 400, including the non-linear pattern ofthe antennas 404-410, causes phase differences within each set 600-606of the antennas, and phase differences across the sets 600-606 ofantennas, and these phase differences collectively give the antennasystem 400 a two-dimensional field of view, whereas a linear array 700(FIGS. 2 and 7 ) has only a one-dimensional field of view.

FIGS. 8-15 illustrate how progressively displacing the antennas 404-410from the longitudinal axis 612 affects the mapping of the segments ofthe beam-side interface 426, specifically the beam ports 428-432, torespective portions (pixels 0, 1, 2, . . . , 15) of the field of view.The displacement is exemplified by arrows 900 and 902 in FIG. 9 andarrows 1000 and 1002 in FIG. 10 . In each of FIGS. 8-15 , the respectiveantenna pattern is shown on the left, and the resulting mapping of beamports 428-432 to the portions (pixels) of the field of view is shown onthe right.

In FIG. 8 , the antennas 404-410 are arranged in a linear pattern, asthe antennas are arranged in FIGS. 1, 2 and 7 , i.e., in the prior art.The antennas 404-410 may be spaced apart by λ/2, where λ is a designwavelength of the antenna array. As a result, each beam port 428-432 ismapped to a band that extends the entire height (Y axis direction) ofthe field of view. In other words, a linear pattern phased array hasresolution in only a dimension parallel to the linear array of antennas.Thus, as noted, a linear phased array is limited to beamforming in asingle dimension, i.e., along the X axis (FIG. 2 ), which corresponds tothe X axis in FIGS. 8-15 .

In each of FIGS. 9-15 , the antennas 404-410 are displaced progressivelyfurther from the longitudinal axis 612. FIG. 9 shows at least anon-empty subset of the antennas of each disjoint set 600-606 displacedperpendicularly a relatively small distance, exemplified bydisplacements 900 and 902, along the second surface from thelongitudinal axis 612. For example, antenna 404 and three other antennasof subset 606 are perpendicularly displaced from the longitudinal axis612. Similarly, antennas of the other subsets 604-600 are alsodisplaced. The antennas 404-410 may be spaced apart along thelongitudinal axis 612 (i.e., parallel to the X axis in FIGS. 9-15 ) anintegral multiple of about one-half the design wavelength λ. As can beseen in the right portion of FIG. 9 , with a relatively small amount ofdisplacement 900 and 902 from the longitudinal axis 612 (i.e., parallelto they axis in FIGS. 9-15 ), each pixel becomes an elongated ellipse,which no longer extends the entire height of the field of view. As shownin FIG. 10 , with further displacement 1000 and 1002, the ellipse ofeach pixel shortens, and each pixel becomes replicated in the field ofview. That is, each beam port 428-432 maps to multiple portions of thefield of view. This progression continues in FIGS. 11-15 , until eachellipse becomes nearly circular in FIG. 15 .

A beam produced by the antenna system 400 has a cross-sectional size,which corresponds to a size of a portion of the field of viewilluminated by the beam, i.e., a size of (angle subtended by) the pixelin the two-dimensional field of view. The cross-sectional size of (anglesubtended by) the beam in a given direction depends on the largestdistance between antennas 404-410 in that direction, i.e., the longestbaseline in that direction. In FIG. 8 , the X direction baseline 800 isapx. 8 wavelengths, and the Y direction baseline is zero. Progressingthrough FIGS. 9-15 , the X direction baseline remains unchanged, whereasthe Y direction baseline progressively increases, up to about 6wavelengths 1502 in FIG. 15 . Thus, the pixel size in the Y directionvaries from about infinity times the size of the pixel in the Xdirection in FIG. 8 , to about 8/6 times the size of the pixel in the Xdirection in FIG. 15 .

Displacing 900, 902, 1000, 1002 the antennas 404-410 from thelongitudinal axis 612 by more than about λ/4 creates grating lobes inthe far field of the Rotman lens 422. These grating lobes form gridsfrom the signals from the antennas 404-410 at points in the field ofview with the same relative distances to the antennas as the main lobe,modulo the wavelength λ. At different wavelengths, the grating lobesappear at slightly different locations in the field of view. FIG. 15shows the main lobes and grating lobes at the center wavelength only,causing them to appear indistinguishable. FIG. 28 shows the same farfield pattern, but at three different wavelengths, represented in red,green, and blue. The main lobes appear white as all three wavelengthsare collocated, while the grating lobes appear in different places foreach wavelength.

As indicated in FIG. 15 , each set, exemplified by set 604, of theantennas 404-410 is tilted at an angle 1500, relative to the pixels inthe far field. Equivalently, if the sets 600-606 of the antennas 404-410of FIG. 15 are not tilted, as shown in FIG. 16 , the pixels are tiltedat the angle 1500. It should be noted that in both FIGS. 15 and 16 ,only one axis is tilted. In other words, the antenna array 402 or thepixel array is a parallelogram, in some cases an equilateral or a nearlyequilateral parallelogram. For an antenna system 400 that includes N(N>1) sets 600-606 of antennas, the slope of the angle 1500 is about1/N.

In practical applications, it may be simpler to fabricate the antennas404-410 in a rectangular array, i.e., without the tilt 1500, as shown inFIG. 16 . In any case, the tilt 1500 should not pose a problem, becausein practical applications, software or hardware that controls drive ofthe antenna system 400 can compensate for the tilt 1500. Furthermore,since the tilt varies with the inverse of N, i.e., 1/N, in practicalapplications with relatively large numbers N of sets 600-606, the tiltis relatively small.

Compact Path-Matching Network

As noted, in some embodiments, the feedlines 412-418 (FIG. 4 ) provideequal effective path lengths between the respective antennas 404-410 andthe beamforming network 420 and/or the Rotman lens 422. In suchembodiments, because some of the antennas 404-410 are physically closerto the beamforming network 420 or the Rotman lens 422 than other of theantennas 404-410, some of the feedlines 412-418 may meander (not shownin FIG. 4 ), while other of the feedlines 412-418 follow straighterpaths, in order to equalize their respective effective path lengthsbetween the respective antennas 404-410 and the beamforming network 420and/or the Rotman lens 422. In some embodiments, physical lengths of therespective paths are equal, in order to achieve equal effective pathlengths. In other embodiments, optionally or alternatively, physicalproperties, such as indexes, of the paths may vary, from path to path,to achieve equal effective path lengths. Here, we focus on meandering asa mechanism to create equal physical path lengths.

A set of feedlines that establishes equal effective path lengths betweenthe antennas 404-410 and the beamforming network 420 or the Rotman lens422 is referred to herein as a “path-matching network.” A path-matchingnetwork typically includes the entire length of each of the feedlines412-418. However, a path-matching network can be thought of as includingonly a portion of the length of each feedline 412-418. In this sense,the path-matching network can be thought of as being communicablycoupled between: (a) ends of the portions of the feedlines 412-418extending to the respective antennas 404-410 and (10) the beamformingnetwork 420 or the Rotman lens 422.

Various meandering arrangements of feedlines may be used to achieveequal effective path lengths. Disclosed herein is a compactpath-matching network. Each such compact path-matching network includesvariations of a meandering arrangement 1700 shown in solid line in FIG.17 . For example, the meandering arrangement 1700 may be repeated, withsuitable adjustments of lengths of individual feedlines, and possiblyleft-to-right and/or top-to-bottom mirroring, for each set 600-606 ofantennas. Although ends of the meandering arrangement 1700 are labeledin FIG. 17 as “To beamformer” and “To antennas,” respectively, the twoends can be interchanged. The meandering arrangement 1700 includes fourexemplary feedlines, representing portions of the feedlines 412-414 thatfeed one set 606 (FIG. 6 ) of antennas 404-406. However, a meanderingarrangement 1700 can include other numbers of feedlines, equal to thenumber of antennas in respective sets of antennas or a subset of a setof the antennas. A portion 1702 of a possible adjacent meanderingarrangement is shown in dashed line in FIG. 17 for reference.

The path-matching network is referred to as “compact,” because adjacentpairs of the feedlines 412-414 are spaced apart from each other aminimum distance, represented by distances 1704, 1706, 1708 and 1710,based on the design wavelength λ of the antenna system 400, taking intoconsideration a desired maximum amount of crosstalk between adjacentfeedlines 412-414, possibly also including a safety margin, andmaterials and fabrication techniques used to make the feedlines 412-414and a substrate on which the feedlines 412-414 are made or disposed.That is, the feedlines 412-414 are as close together as possible,without incurring undue cross talk between the feedlines 412-414. Insome embodiments, the spacing 1704-1710 is as little as a fewwavelengths λ.

Width 1712 of the meandering arrangement 1700 means lateral spacerequired for the feedlines 412-414, including space 1710 between themeandering arrangement 1700 and the adjacent meandering arrangement 1702on one side. (“Lateral” in this context means perpendicular to afeedline 412-414 at any given point along the feedline, or perpendicularto a general direction of a feedline 412-414.) In other embodiments, apath-matching network may include the meandering arrangement describedherein, although with wider than minimum spacings 1704-1710.

In the meandering arrangement 1700 shown in FIG. 17 , each feedline412-414 includes two bends, exemplified by first bends 1714 and 1716 infeedlines 412 and 414, respectively, and second bends, exemplified bybends 1718 and 1720, in feedlines 412 and 414, respectively. In othermeandering arrangements (not shown), each feedline may include more thantwo bends. A reverse curve includes two bends in opposite directions.

Among the feedlines 412-414, the first bends 1714-1716 vary in radius,and the second bends 1718-1720 vary in radius. For example, in feedline412, the first bend 1714 has a larger radius than the first bend 1716 infeedline 414. Intermediate feedlines have intermediate radii bends.Tighter bends tend to induce more waveguide losses than wider bends.Therefore, to an extent practical, each feedline 412-414 should have anintegrated curvature approximately equal to the integrated curvature ofeach other feedline 412-414, to approximately equalize the losses in thefeedlines 412-414. For example, in feedline 412, the first bend 1714 isrelatively wide and the second bend 1718 is relatively tight, whereas infeedline 414 the first bend 1716 is relatively tight and the second bend1720 is relatively wide, but the sum of the radii 1714 and 1718 at leastapproximately equal the sum of the radii 1716 and 1720. Similarly, thesum of the radii of each intermediate feedline at least approximatelyequals the sum of radii 1714 and 1718, and at least approximately equalsthe sum of radii 1716 and 1720.

All the feedlines 412-414 of the meandering arrangement 1700 shown inFIG. 17 have equal lengths and, therefore, equal effective path lengths,assuming substantially the same materials are used to fabricate each ofthe feedlines 412-414. However, as noted, some of the antennas of theset 606 (FIG. 6 ), such as antenna 404, are closer to the beamformingnetwork 420 (FIG. 4 ) or the Rotman lens 422 than other of the antennas,such as antenna 406. Thus, some of the feedlines 412-414 in FIG. 17 ,such as the feedline 412, should be made longer than the otherfeedlines, such as feedline 414, to equalize the effective path lengthsto the respective antennas 404-406.

FIG. 18 shows, in dashed line, how each of the feedlines 412-414 of FIG.17 can be lengthened, without increasing the width 1712 of themeandering arrangement 1700. Although all the feedlines 412-414 arelengthened by equal amounts (two times distance 1800) in FIG. 18 , eachfeedline 412-414 can be lengthened a different amount, as needed, forexample as indicated at 1900, 1902, 1904 and 1906 in FIG. 19 .

With the meandering arrangement 1700 shown in FIGS. 17-19 , eachfeedline 414, etc. can be lengthened no more than the feedline to itsleft. Thus, delays introduced by lengthening the feedlines 412-414increase monotonically from the right-most feedline 414 to the left-mostfeedline 412. If, however, it is necessary to monotonically increasedelays from left to right, the meandering arrangement 1700 can bemirrored left-to-right, as shown in FIG. 20 .

FIG. 21 illustrates dimensions, coordinates and mathematicalrelationships there among for designing a compact path-matching networkof N feedlines 412-414, according to an embodiment of the presentinvention. Portions of some of the feedlines 412-414 are omitted fromFIG. 21 for clarity.

As noted, the feedlines 412-414 are spaced apart by spacings 1704-1710.In general, the feedlines 412-414 are parallel to each other. However,inter-feedline spacing need not necessarily be equal along the entirelengths of a feedlines 412-414, particularly along portions of thefeedlines 412-414 where the feedlines 412-414 are lengthened bydifferent amounts, as discussed with respect to FIG. 19 , for example inan area 1908.

A smallest radius R_(S) of curvature of the feedlines 412-414 isarbitrary, in that R_(S) does not depend on any of the othermeasurements shown in FIG. 21 . The smallest radius R_(S) of curvaturemay, for example, be selected based on a desired maximum bending loss(insertion loss) of the respective curved portions of the feedline412-414. Other things being equal, bending loss generally increases withdecreased bending radius. Bending loss can be calculated or estimated bywell-known methods, for example based on bending radius, waveguide size,wavelength, waveguide materials used and fabrication method.

Considerations for selecting the inter-feedline spacing 1704-1710 arediscussed herein, with reference to FIG. 17 .

A largest radius R_(L) of curvature of the feedlines 412-414 depends onR_(S), the inter-feedline spacing 1704-1710 and the number N offeedlines 412-414, according to equation (1):R _(L) =R _(S)+(N−1)×Spacing  (1)

Each feedline 412-414 includes straight sections, exemplified bystraight sections 2100, 2102 and 2104 (FIG. 21 ), and curved sections,exemplified by curved sections 2106 and 2108. Each curved section 2106and 2108 is a portion of a circle. Curved section 2108 is a semicircle,and curved section 2106 includes a semicircle and an S-shaped (reversecurve) portion 2110. A reverse curve is a curve to the left or rightfollowed immediately by a curve in the opposite direction. The S-shapedportion 2110 is also shown separately in an insert box 2112, forclarity. As can be seen most clearly in the insert box 2112, theS-shaped portion 2110 includes two opposite curves 2114 and 2116,indicated by dotted and dashed lines, respectively. Each of the twoopposite curves 2114 and 2116 is part of a respective circle indicatedby dotted lines 2118 and 2120, respectively. The two opposite curves2114 and 2116 are joined, end-to-end, such that the two opposite curves2114 and 2116 are tangent to each other.

The curved section 2106 and the corresponding curved sections of theother feedlines 414, etc., including dotted circle 2118, are allcentered on a first common point 2122. FIG. 21 shows radii R_(S) andR_(L) extending from the first common point 2122.

The curved section 2110 and the corresponding curved sections of theother feedlines 414, etc., including dotted circle 2120, are allcentered on a second common point 2124, distinct from the first commonpoint 2122. Additional dotted circles are centered on the second commonpoint 2124, and the corresponding curved sections of the other feedlines414, etc., follow respective portions of the additional dotted circles.

If the first common point 2122 is thought of as being positioned atcoordinates (0, 0), coordinates (x, y) of the second common point 2124can be calculated as follows. A portion 2126 of the feedline 414 followsa dotted circle 2128, which has a radius R_(S) and is centered at thesecond common point 2124. The dotted circle 2128 is tangent to dashedcircle 2130, which is centered at the first common point 2122 and hasradius R_(L). Thus, the dotted circle 2128 must be centered on dashedcircle 2132, centered at the first common point 2122 and having a radiusR_(L)+R_(S). The x coordinate of the second common point 2124 iscalculated according to equation (2):x=−N×Spacing  (2)

Equation (2) is explained with reference to FIG. 22 , which shows anenlarged portion of FIG. 21 . Consider a horizontal line 2200 extendingfrom the left-most feedline 412 to the second common point 2124. Sincethe first common point 2122 is at the origin (0, 0), and the smallestdotted circle 2118 has radius R_(S), a point of tangency 2202 of thefeedline 412 and the smallest dotted circle 2118 has an x coordinateequal to R_(S). Thus, the x coordinate of point 2204, where the line2200 intersects the (vertical) feedline 412 is also R_(S). From thepoint 2204, the line 2200 extends left to the second common point 2124.The line 2200 intersects the smallest dotted circle 2128 at a point2206. The distance between the points 2204 and 2206 is N times thespacing 1706. Thus, the x coordinate of the point 2206 isR_(S)−(N×Spacing). Since the second common point 2124 is R_(S) units tothe left of the point 2206, the x coordinate of the second common point2124 is R_(S)−(N×Spacing)−R_(S)=−(N×Spacing), as shown in equation (2).

As noted, the dashed circle 2132 is centered on coordinates (0, 0). Theequation for the dashed circle 2132 is:x ² +y ²=(R _(L) +R _(S))²  (3)

Solving equation (3) for y negative:

$\begin{matrix}{y = {- \sqrt{\left( {R_{L} + R_{S}} \right)^{2} - x^{2}}}} & (4)\end{matrix}$

Substituting x from equation (2) into equation (4):

$\begin{matrix}{y = {- \sqrt{\left( {R_{L} + R_{S}} \right)^{2} - \left( {N \times {Spacing}} \right)^{2}}}} & (5)\end{matrix}$

Thus, equations (2) and (5) give the (x, y) coordinates of the secondcommon point 2124, relative to the first common point 2122.

Although the path lengths in FIG. 17 are equal, more tightly curvedpaths incur more signal losses than less tightly curved paths. Thus, forexample, although they are of equal length, feedline 412 may suffer moresignal loss than feedline 414, other things being equal. FIG. 45illustrates two variations 4500 and 4502 of the meandering arrangement1700 of FIG. 17 that solve this problem. In the meandering arrangement4500 shown in FIG. 45 , each curve (shown in dashed line) is identical,including having an identical radius, exemplified by radii 4504 and4506. Thus, feedlines 4508 and 4510 are of equal length and incur equalsignal losses due to having identical curves.

A meandering arrangement can include more than one type (size and/orshape) curve. For example, meandering arrangement 4502 includes twotypes of curves, i.e., quarter circles in region 4512, and ⅛ circles inregion 4514. However, within each region 4512-4514, respective curvesare identical.

The curves shown in FIG. 45 are portions of circles. However, the curvesneed not necessarily be portions of circles. Other shaped curves (notshown) may be used, as long as they are identical, within manufacturingtolerances, within respective regions.

Narrowband Two-Dimensional Planar and Crossover-Free Beamforming NetworkArchitecture

Returning to FIG. 4 , assuming a suitable path-matching network, asdescribed with reference to FIGS. 17-22 and 45 , is included, and theantennas 404-410 are suitably spaced apart horizontally and vertically,the antenna system 400 has a two-dimensional field of view, as discussedwith reference to FIGS. 5 and 9-16 , yet can be implemented on asurface, such as on an electronic or photonic integrated circuit, withno waveguide crossing. As described thus far, the antenna system 400 hasa relatively narrow usable bandwidth outside of a central broadband rowof pixels, in that the direction of the beam changes with wavelength,causing bandwidth spreading, which worsens with distance from thecentral row. For example, for an N×N device, a p % change in wavelengthof a signal fed to a given beam-side interface port 428-432 causes achange in direction of the beam equal to about N×p % of the beam size(effective pixel size) per row away from the center. Consequently, thespectrum of each pixel of a broadband signal is somewhat “smeared” inthe field of view, similarly to white light through a prism.

Resolution in a given dimension is inversely proportional to maximumseparation of antennas in that dimension. If the maximum separation ofantennas is too small, there will be insufficient resolution todifferentiate along that dimension, causing the antenna array toeffectively have a one-dimensional field of view. In addition, the sizeof the grating-lobe-free field of view (the size of box 502 in FIG. 5 ,or the areas with only white spots in FIGS. 38-41 ) in a given dimensionis inversely proportional to the minimum separation of antennas in thatdimension. Therefore, spreading the antennas out in a given dimensionshrinks the field of view and increases the number of grating lobes(i.e., replicated ellipses), which could be useful in some cases andproblematic in others, depending on design objectives of the antennasystem.

The antenna system 400 of FIG. 4 can be made usable over a widebandwidth by modifying the two-dimensional beamforming network 420, forexample as shown in FIG. 23 . FIG. 23 is a diagram of a crossover-freeantenna system 2300, similar to the antenna system 400 of FIG. 4 .However, the two-dimensional beamforming network 420 includes threestages 2302, 2304 and 2306 of beamforming networks. The first stage 2302includes N first beamforming networks, exemplified by first beamformingnetworks 2308, 2310 and 2312. The second stage 2304 includes N secondbeamforming networks, exemplified by second beamforming networks 2314,2316 and 2318. The third stage 2306 includes a third beamforming network2320.

Each first beamforming network 2308-2312 is associated with a distinctset of the antennas 404-410. For example, in the embodiment shown inFIG. 23 , continuing with the example antenna sets 600-606 of FIG. 6 ,first beamforming network 2308 is associated with set 606 of theantennas 404-410, and first beamforming network 2310 is associated withset 604 of the antennas 404-410. Each first beamforming network2308-2312 has a beam-side interface, exemplified by beam-side interfaces2322, 2324 and 2326.

Each first beamforming network 2308-2312 also has a plurality ofarray-side ports. For example, first beamforming network 2308 hasarray-side port 2328, and first beamforming network 2312 has array-sideports 2330, 2332 and 2334. The array-side ports 2328-2334 of each firstbeamforming network 2308-2312 are individually communicably coupled torespective antennas 404-410 of the associated set 600-606 of theantennas 404-410. For example, the array-side port 2328 is individuallycommunicably coupled to antenna 404 via feedline 412. (The earlierdiscussion, with reference to FIG. 4 , of individually communicablycoupled antennas 404-410 also applies to the embodiment shown in FIG. 23.) The array-side ports 2328-2334 of the N first beamforming networks2308-2312 thereby collectively form the one-dimensional array-sideinterface 424 of the two-dimensional beamforming network 420.

Each second beamforming network 2314-2318 is associated with a distinctfirst beamforming network 2308-2312. For example, second beamformingnetwork 2314 is associated with first beamforming network 2308, andsecond beamforming network 2316 is associated with first beamformingnetwork 2310.

Each second beamforming network 2314-2318 has a beam-side interface,exemplified by beam-side interfaces 2336, 2338 and 2340. Each beam-sideinterface 2336-2340 of each second beamforming network 2314-2318 iscommunicably coupled to the beam-side interface 2322-2326 of itsassociated first beamforming network 2308-2312. For example, beam-sideinterface 2336 of second beamforming network 2314 is communicablycoupled to beam-side interface 2322 of first beamforming network 2308.In some embodiments, each beam-side interface 2336-2340 of each secondbeamforming networks 2314-2318 is communicably coupled to the associatedbeam-side interface 2322-2326 of the associated first beamformingnetwork 2308-2312 via a respective group of feedlines, exemplified byfeedline groups 2342 and 2344. However, in some embodiments, describedherein, a bulk medium, rather than individual feedlines, communicablycouples each beam-side interface 2336-2350 to the associated beam-sideinterface 2322-2326 of the associated first beamforming network2308-2312.

Each second beamforming network 2314-2318 has an array-side interface,exemplified by array-side interfaces 2346, 2348 and 2350. The thirdbeamforming network 2320 has an array-side interface 2352. Thearray-side interface 2346-2350 of each second beamforming network2314-2318 is communicably coupled to a respective distinct portion ofthe array-side interface 2352 of the third beamforming network 2320. Forexample, array-side interface 2348 of second beamforming network 2316 iscommunicably coupled to a portion 2354 of the array-side interface 2352of the third beamforming network 2320.

The third beamforming network 2320 has a plurality of beam-side ports,exemplified by beam-side ports 2356, 2358 and 2360. The plurality ofbeam-side ports 2356-2360 of the third beamforming network 2320collectively forms the segmented one-dimensional beam-side interface 426of the two-dimensional beamforming network 420.

Each first beamforming network 2308-2312 may consist essentially of arespective one-dimensional beamforming network. The first beamformingnetworks 2308-2312 may all be disposed on a common surface. In theembodiment shown in FIG. 23 , each first beamforming network 2308-2312includes a single respective Rotman lens, represented by Rotman lenses2362 and 2364. Each second beamforming network 2314-2318 may consistessentially of a respective one-dimensional beamforming network,distinct from the first beamforming networks 2308-2312. The secondbeamforming network 2314-2318 may all be disposed on a common surface,which can, but need not, be the same surface on which the firstbeamforming networks 2308-2312 are disposed. In the embodiment shown inFIG. 23 , each second beamforming network 2314-2318 includes a singlerespective Rotman lens, represented by Rotman lenses 2366 and 2368. Thethird beamforming network 2320 may consist essentially of a distinctone-dimensional beamforming network, distinct from the first and secondbeamforming networks 2308-2312 and 2314-2318. The third beamformingnetwork 2320 may, but need not, be disposed on the same surface as thefirst and second beamforming networks 2308-2312 and 2314-2318. In theembodiment shown in FIG. 23 , the third beamforming network 2320includes the single Rotman lens 422.

In another embodiment (not shown), the Rotman lenses of the firstbeamforming network 2308-2312 are reused for the second beamformingnetworks 2314-2318 by essentially folding the device in half across themiddle to overlap the Rotman lenses. A 180 degree turn (fold) can beimplemented with a mirror. The path delays between the first and secondlayers (of the broadband Fourier lens) then become longer or shorterpaths to the mirrors. A Rotman lens used both forwards and backwards,combined with monotonically increasing paths with mirrors on the ends,is known as a Rotman-lens spectrum decomposer (RLSD), an example ofwhich is described in FIG. 11 of X. Wang, et al., “Flexible-Resolution,Arbitrary-Input, and Tunable Rotman Lens Spectrum Decomposer,” IEEETransactions on Antennas and Propagation, vol. 66, no. 8, pp. 3936-3947,August 2018, doi: 10.1109/TAP.2018.2839896 (“Wang”), the entire contentsof which are hereby incorporated by reference herein, for all purposes.The array side interfaces of a combined/folded Rotman lens correspond tothe ports on the left side of Wang's FIG. 11 , and the beam sideinterfaces, i.e., where the device is folded, are on the right side ofWang's FIG. 11 . An RLSD-based embodiment may not necessarily bemanufacturable on a single flat surface without crossovers or extraassembly steps. However, RLSDs implemented on multilayer photonic chipsmay be used.

Returning to the embodiment shown in FIG. 23 , two additional Rotmanlenses, for example Rotman lenses 2362 and 2366, are interposed betweeneach set 600-606 of antennas and the “primary” Rotman lens 422.Inserting this pair of Rotman lenses 2362 and 2366 effectively reversesthe longitudinal order of the feedlines at 2352, relative to theirordering at 424, within each set 600-606 of antennas, because a Fouriertransform of a Fourier transform is effectively an order reversal.“Longitudinal” in this context means in a direction parallel to thelongitudinal axis 612.

Reversing the longitudinal order of the feedlines changes the order ofthe pixels in the field of view of the antenna system 2300, asillustrated in FIG. 24 . A simplified version of the antenna system 400is shown on the top-left portion of FIG. 24 , along with its far fieldpixel mapping in the bottom-left portion of the figure. For simplicityof explanation, FIG. 24 shows only two groups 602 and 606 of antennas,and only two antennas, exemplified by antennas 410 and 406, in eachgroup. Like reference numerals in FIGS. 4, 23 and 24 refer to likeitems.

A simplified version of the antenna system 2300 is shown in the middle(left to right) of FIG. 24 , along with its far field pixel mapping. Asshown on the right in FIG. 24 , reversing the order of the feedlines ineach set 600-606 of antennas counteracts the reversal due to the pair ofRotman lenses 2362 and 2366. Note that the far field pixel mapping onthe right in FIG. 24 is the same as the far field pixel mapping on theleft in FIG. 24 , whereas the pixel mapping in the center in FIG. 24 isdifferent from that shown in the left and right in FIG. 24 . Thus,applying this feedline reversal scheme to the antenna system 2300 ofFIG. 23 , the array-side ports 2328-2334 of the first beamformingnetwork 2302, 2308-2312 would be transversely ordered in a first order.The antennas 404-410 of the associated set of the antennas would betransversely ordered in a second order, and the antennas of theassociated set of the antennas would be individually communicablycoupled to the respective array-side ports 2328-2334 such that the firstorder is opposite the second order.

FIG. 25 is a diagram of a crossover-free antenna system 2500, similar tothe antenna system 2300 of FIG. 23 , however in FIG. 25 , the feedlinesin each set 600-606 of antennas are longitudinally reversed, asdiscussed with respect to FIG. 24 . Differences in the lengths of thefeedlines 412-418, as a result of reversing their order, can becompensated with the path-matching network described with reference toFIGS. 17-22 . Assuming a suitable path-matching network is included, andthe antennas 404-410 are suitably spaced apart horizontally andvertically, the antenna system 2500 has a two-dimensional field of view,as discussed with reference to FIGS. 5 and 9-16 , yet can be implementedon a surface, such as on an electronic or photonic integrated circuit.The antenna system 2500 has a relatively narrow useable bandwidth, asdiscussed with respect to the antenna system 400 of FIG. 4 . The rightportion of FIG. 24 is repeated in the left portion of FIG. 26 , exceptin FIG. 26 the far field pixel mapping is plotted in broadband, ratherthan narrowband, as it is in FIG. 24 . A color/hash code 2600 is used inFIG. 26 to show the spectral smearing of some of the pixels.

Main lobes from the two-dimensional beamformer 420 appear white, whileside lobes smear out into rainbows. As can be seen in the left portionof FIG. 26 , central pixels 0, 1, 2 and 3 are not spectrally smeared,i.e., these pixels are broadband pixels. However, these central pixels0-3 provide directivity and differentiation of signals in only onedimension, i.e., along the X axis.

Broadband Two-Dimensional Planar and Crossover-Free Beamforming NetworkArchitecture

As shown in the center and right portions of FIG. 26 , as contrastedwith the left portion of FIG. 26 , embodiments of the present inventionrearrange the pixels in the far field pixel mapping to providetwo-dimensional broadband beamforming by introducing particular delayrelationships among the feedlines 2342, etc., as represented by feedline2602 being longer (incurring more delay) than feedline 2604. Althoughthe feedline 2604 is the same length (incurs the same delay) in theright and center portions of FIG. 26 , the feedline 2602 is longer(incurs more delay) in the right portion of FIG. 26 than in the centerportion of FIG. 26 . Thus, the difference between the delays incurred bythe feedlines 2602 and 2604 is greater in the right portion of FIG. 26than in the center portion of FIG. 26 . Note that the broadband pixels0, 1, 2 and 3 are arranged in a square grid array in the right portionof FIG. 26 , whereas these pixels 0, 1, 2 and 3 are arranged inapproximately a parallelogram grid array in the center portion of FIG.26 .

As noted, the delay relationships involve particular differences indelays in the various feedlines within each set 600-606 of antennas andacross the sets 600-606 of antennas. However, as can be seen bycomparing the center and right portions of FIG. 26 , increasing, toappropriate values, differences between the least delayed and mostdelayed individual feedlines for each set 600-606 of antennas, anddifferences between the least and most delayed sets 600-606, as well asappropriately adjusting the delays in intermediate feedlines and sets(none shown in FIG. 26 ), results in a two-dimensional broadband arrayof pixels in the center of the field of view. These delays shift themain lobes into the grid, enabling broadband imaging in the center ofthe far field.

Referring back to FIG. 25 , each group of feedlines 2342-2344 can beconsidered a respective column 2502, 2504, 2506 and 2508 of feedlines,even if the individual feedlines are not straight. The delays areintroduced within each column 2502-2508 and across the columns2502-2508.

Each column 2502-2508 corresponds to a respective group of antennas600-606. Each group of antennas 600-606 is disposed a respective lateraldistance from a center line 2510. In FIG. 25 , the groups of antennas600-606 are assumed to be laterally equally spaced apart. However, insome embodiments, the groups of antennas 600-606 are unequally laterallyspaced apart. That is, lateral spacings between adjacent groups ofantennas 600-606 need not necessarily all be equal to each other. Insome cases, entire groups of antennas or the antennas within the groupsmay be shifted, when coupled with appropriate changes to the delay lineslengths, or they may be omitted entirely. The number of groups ofantennas does not necessarily need to be even. For example, inembodiments with odd numbers of groups of antennas 600-606, a centergroup of the antennas 404-410 may be disposed a zero lateral distancefrom the center line 2510.

We now discuss an exemplary case of evenly spaced columns without shiftsor omissions. If the two-dimensional beamforming network 420 contains anodd number of columns 2502-2508 (odd number of groups of antennas600-606), the columns 2502-2508 can be monotonically laterally numberedwith unique integers (j) according to equation (6):

$\begin{matrix}{j \in {{{\mathbb{Z}}: - \left\lfloor \frac{N}{2} \right\rfloor} \leq j \leq {+ \left\lfloor \frac{N}{2} \right\rfloor}}} & (6)\end{matrix}$where N is the number of groups of antennas 600-606 (equal to the numberof columns 2502-2508). For example, if N=7, the columns 2502-2508 wouldbe numbered, in order: −3, −2, −1, 0, +1, +2 and +3. Essentially, forany given column 2502-2508, j equals the number of columns thecorresponding group of antennas 600-606 is laterally disposed from thecenter line 2510.

On the other hand, if the two-dimensional beamforming network 420contains an even number of columns 2502-2508 (even number of groups ofantennas 600-606), the columns 2502-2508 can be monotonically laterallynumbered with unique rational numbers (j), spaced apart by 1, accordingto equation (7):

$\begin{matrix}{j \in {{{{\mathbb{Q}}: - \left\lfloor \frac{N}{2} \right\rfloor} + \frac{1}{2}} \leq j \leq {{+ \left\lfloor \frac{N}{2} \right\rfloor} - \frac{1}{2}}}} & (7)\end{matrix}$For example, if N=4, the columns 2502-2508 would be numbered, in order:−1.5, −0.5, +0.5 and +1.5.

Alternatively, the columns 2502-2508 can be monotonically laterallynumbered with unique rational numbers (j), spaced apart by 1, accordingto equation (8):

$\begin{matrix}{j \in {{{{\mathbb{Q}}:} - \left\lfloor \frac{\left( {N - 1} \right)}{2} \right\rfloor} \leq j \leq {+ \left\lfloor \frac{\left( {N - 1} \right)}{2} \right\rfloor}}} & (8)\end{matrix}$

Alternatively, the columns 2502-2508 can be monotonically laterallynumbered with integers 0 to (N−1), then subtract (N−1)/2 from eachnumber.

Assuming the array of antennas 402 is an N×N array, each column containsN feedlines, and the feedlines in each column can be numbered using thesame scheme. Thus, for example, if N=4, the feedlines in each column arenumbered, in order: −1.5, −0.5, +0.5 and +1.5. Essentially, for anygiven column 2502-2508, j equals the number of columns the correspondinggroup of antennas 600-606 is laterally disposed (displaced) from thecenter line 2510, as shown in FIG. 25 . Note that if N is even, no setof antennas 606-606 is disposed on the center line 2510, so each set ofantennas 600-606 is laterally disposed an odd integral multiple of ½columns from the center line 2510.

In one embodiment, the maximum difference in delay (referred to hereinas a “scaling factor”) in the two-dimensional beamforming network 420,i.e., the delays between the first stage 2302 and the second stage 2304of the two-dimensional beamforming network 420, not including internaldelays of the Rotman lenses 2362-2368, is calculated according toequation (9):

$\begin{matrix}{{{Scaling}{factor}} = {2 \times \left( \frac{N - 1}{2} \right)^{2}}} & (9)\end{matrix}$assuming the array of antennas 402 is an N×N array. That is, thefeedline having the least delay and the feedline having the greatestdelay differ in delay amount by the scaling factor. Using the 4×4 arrayof antennas 402 shown in FIG. 25 as an example (N=4), the scaling factoris 4.5. The scaling factor is in units of design wavelengths λ.

More generally, for an M×N rhombus array, we have:

$\begin{matrix}{{{Scaling}{factor}} = {2 \times \left( \frac{N - 1}{2} \right) \times \left( \frac{M - 1}{2} \right)}} & (10)\end{matrix}$

To convert this into a square device, i.e., to make both the antenna andthe far field pixel arrays rectangular, multiply the delay lengths by acorrection factor of

(1 + 1/N²),where Nis the number of rows/number of ports on the smaller Rotmanlenses.

Thus, a more general equation for the scaling factor for an N×M device(N rows, M columns) is:

$\begin{matrix}{{{Scaling}{factor}} = {2 \times \left( \frac{N - 1}{2} \right) \times \left( \frac{M - 1}{2} \right) \times \left( {1 + \frac{1}{N^{2}}} \right)}} & (11)\end{matrix}$

Thus, for example, a 4×5 square device (4 rows and 5 columns) has amaximum delay difference of 6.375.

For each column j 2502-2508, relative delays among the feedlines of thecolumn vary from:

$\begin{matrix}{{- \left( \frac{N - 1}{2} \right)}j\lambda} & (12)\end{matrix}$

$\begin{matrix}{{+ \left( \frac{N - 1}{2} \right)}j{\lambda.}} & (13)\end{matrix}$

For example, using the 4×4 antenna array 402 shown in FIG. 25 , therelative delays among the feedlines in the first column (column number−1.5) of feedlines vary from −2.25 to +2.25, i.e., a maximum differenceof 4.5, which is consistent with the above-calculated scaling factor.

Continuing with the 4×4 array of antennas 402 of in FIG. 25 , Table 1lists a relative amount of delay in each feedline between the respectivefirst and second stages 2302 and 2304 (first beamforming networks2308-2312 (FIG. 25 ) and second beamforming networks 2314-2318), withpositive numbers corresponding to effectively longer paths than negativenumbers. Columns of Table 1 correspond to columns 2502-2508 (ex., Columnnumber −1.5, Column number −0.5, etc.) of feedlines in thetwo-dimensional beamforming network 420, and rows of Table 1 correspondto individual feedlines (ex., Feedline number −1.5, Feedline number−0.5, etc.) of a given column 2502-2508.

TABLE 1 Relative Path Delays Between First and Second Stage BFNs Column−1.5 Column −0.5 Column +0.5 Column +1.5 Feedline −1.5 −2.25 −0.75 +0.75+2.25 Feedline −0.5 −0.75 −0.25 +0.25 +0.75 Feedline +0.5 +0.75 +0.25−0.25 −0.75 Feedline +1.5 +2.25 +0.75 −0.75 −2.25

Corner values in Table 1 may be calculated according to equations (12)and (13). Intermediate values in the top and bottom rows, and in theleft and right columns, are evenly distributed between the values in theends of the row or column, assuming the antennas 404-410, of the arrayof antennas 402, are regularly spaced apart. Each central sub-square ofTable 1 can be similarly calculated, as though the sub-square representsa smaller M×M/array. For example, the sub-square that consists of fourcentral cells (−0.5 to +0.5×−0.5 to +0.5) can be treated as though itrepresents a 2×2 array.

In one embodiment, if N is odd, and the central set 600-606 of antennas404-410 is centered on the center line 2510 (FIG. 25 ), the center rowof Table 1 and the center column of Table 1 contain all zeros.

Corners of Table 1 may also be calculated using the scale factor, sinceopposite ends of the top and bottom rows, and opposite ends of the leftand right columns have differences equal to the scale factor.

Note that cells in Table 1 contain relative delay amounts, among thefeedlines between the respective first and second stages 2302 and 2304.Clearly, negative absolute delay amounts are physically impossible toimplement. However, using the meandering arrangement 1700 describedherein with reference to FIGS. 17-20 , any desired monotonicallyincreasing relative delays can be implemented laterally across eachcolumn 2502-2508 of feedlines, for example as shown in FIG. 27 . In thiscontext, “laterally” means across one of the columns 2502-2508 or acrossthe entire set of columns 2502-2508, left-to-right or right-to-left (asthe case may be), for example as indicated by double-headed arrow 2700.“Monotonically” has its ordinary mathematical meaning.

Feedlines with the smallest path delays (for example, −2.25 in Table 1)are shortest, and feedlines with greater path delays (for example,Feedline +1.5 in Column +0.5, with a relative delay of −0.75) are madesuitably longer, so as to impose more delay, such that the relativedelays in the feedlines are as prescribed above. Thus, the relativedelay amounts shown in Table 1 may be implemented with a meanderingarrangement 1700, along the lines shown in FIG. 27 .

As an additional example, FIG. 43 contains Table 2 of relative pathdelays between first and second stage beamforming networks for anexemplary broadband 7×7 antenna array (not shown).

Thus, in an antenna system with a design wavelength k, the beam-sideinterface (2336-2340) of each second beamforming network (2304,2314-2318) may be communicably coupled to the beam-side interface(2322-2326) of the associated first beamforming network (2302,2308-2312) by a respective associated first coupling (2342-2344,2502-2508), thereby collectively defining a plurality of firstcouplings. Each non-central first coupling (2342-2344, 2502-2508) isconfigured to delay signals of wavelength λ propagating therethrough bya respective relative delay amount. The delay amount variesmonotonically transversely across the non-central first coupling(2342-2344, 2502-2508).

FIGS. 28-34 illustrate how progressively increasing the relative delaysin the feedlines of the columns 2502-2508 progressively shifts thepixels from a one-dimensional beamforming arrangement (when operating inbroadband), as discussed with respect to the left portion of FIG. 26 ,to a two-dimensional beamforming arrangement. In each of FIGS. 28-34 ,relative delays in respective feedlines are represented by heights ofbars in a bar graph on the left, and a resulting pixel mapping is shownon the right. The color/hash code 2600 of FIG. 26 also applies to theright portions of FIGS. 28-34 . FIGS. 28-34 were generated based on red(644 nm), green (560 nm) and blue (476 nm) signals, with greenrepresenting the design wavelength, and red and blue representing thedesign wavelength +15% and −15%, respectively.

As noted, for an N×N device, a p % change in wavelength of a signal fedto a given beam-side interface port 428-432 (FIG. 4 ) causes a change indirection of the beam equal to about N×p % of the beam size (effectivepixel size) per row moved away from the center. Consequently, thespectrum of each pixel of a broadband signal is somewhat “smeared” inthe field of view. For example, for a 5% bandwidth, the smearing is 5%per row moved away from the central broadband row. For example, in FIG.28 , the row 2800 of pixels located two rows above the central row 2802,and the row 2804 located two rows below the central row 2802, i.e.,±0.0.25 along the sin(Y Angle) axis, have 10% bandwidth smearing. Thepercentage is a ratio of the size of the grating lobe-free field of view(size of box 502 in FIG. 5 ), which means the smearing is about N timesthe spot size. The rows also shift left by one field-of-view per rowmoved upwards, and one field-of-view right per row downwards. Followinga given pixel, as we move up a row, the movement of the pixel is mostlyleftwards, offset by angle 1500 (FIG. 15 ). This direction is also thedirection of the smearing, as can be seen in the spots at (−0.25, 0.5)and (−0.75, 0.5) in the left-most image of FIG. 26 .

Pixels that move in to fill in gaps caused by the leftward shifting rowsas we move up from the central row act a bit differently, as can be seenin the spots at (0.25, 0.5) and (0.75, 0.5) in the left-most image ofFIG. 26 , but the amount of smearing is similar and in a differentdirection.

In FIG. 28 , all the feedlines have identical delays, i.e., zerorelative delay across all the feedlines in the feedline groups2342-2344. In FIG. 29 , bars, represented by bars 2900, 2902 and 2904,represent respective relative delays in the feedlines. Each group 2906,2908, 2910 and 2912 of bars 2900-2904 corresponds to a respective column2502-2508 of feedlines (FIG. 27 ). Bars 2900-2904 extending up from thehorizontal axis represent positive relative delays, and bars 2900-2904extending down from the horizontal axis represent negative relativedelays. As noted, since negative real delays are impossible toimplement, all the feedlines can be made appropriately longer, so as tointroduce the relative delays shown. As can be seen, for example inFIGS. 32-34 , with appropriate delays, as calculated according toequations (7) to (13), a square pixel grid can be achieved.

Operation of the antenna system 400 may be described as follows. EachRotman lens 2362-2364 (FIG. 23 ) in the first stage 2302 selects aparticular delay line in its respective group of feedlines 2342-2344,based on an angle of incidence at which a plane wave impinges on thearray of antennas 402, along the column (vertical) direction. All signalfrom a particular column goes through that delay line and incurs acorresponding delay. Each Rotman lens 2366-2368 in the second stage 2304separates the signal from that delay line back out and spreads thesignal over its set of waveguides. The lengths of the delay lines differfor each column, such that a phase slope is imposed between columns. TheRotman lens 422 in the third stage 2306 takes all of the inputs andfocuses them to a port, based on the total slope.

Parallelogram Antenna Array and/or Pixel Array

As noted, a parallelogram layout of antennas can pose manufacturingchallenges. The left portion of FIG. 35 illustrates a simplified versionof the antenna system 400 and its corresponding pixel map, similar tothe right portion of FIG. 24 , except the antenna system in FIG. 35includes the relative path delays described herein in the feedlines2602-2604. Consequently, the antenna system of FIG. 35 operates inbroadband. The color/hash code 2600 of FIG. 26 also applies to FIG. 35 .

The left portion of FIG. 35 illustrates a parallelogram layout ofantennas, which may pose challenges. For example, if a lens array (notshown) is disposed between the antennas 406-410 and the far field, acustom lens array may be required. As noted, the antennas 406-410 can beshifted to form a square grid, as shown in the right portion of FIG. 35. The center portion of FIG. 35 shows an intermediate shift of theantennas 406-410. Although a square grid of antennas 406-410 allows useof a more conventional lens array, the square antenna grid distorts thebeam pattern into a parallelogram by tilting the horizontal dimension toa slope of about 1/N.

However, as noted, larger values of N cause less slope, as illustratedby comparing FIG. 35 and FIGS. 36-37 . FIG. 36 is similar to FIG. 15 ,and FIG. 37 is similar to FIG. 16 , except that the antenna system 400represented in FIGS. 36 and 37 includes the relative path delaysdescribed herein in the feedlines 2602-2604. The color/hash code 2600 ofFIG. 26 also applies to FIGS. 36-37 . The value of N in FIGS. 36 and 37is greater than the value of N in FIG. 35 . Consequently, the slope 3700(FIG. 37 ) is less than the slope 3500 (FIG. 35 ).

Antenna Spacing and Pixel Size

Spacings between the antennas of the antenna system 400 affect the sizesof the pixels, as illustrated in FIG. 38 . The left portion of FIG. 38is similar to the right portion of FIG. 35 , and the color/hash code2600 of FIG. 26 applies to FIG. 38 . The antennas of the antenna system400 are progressively closer together in the center and right portionsof FIG. 38 . As can be seen, the sizes of the pixels are progressivelylarger in the center and right portions of FIG. 38 .

The relationship between antenna spacing and pixel size is furtherillustrated for a larger antenna system 400 in FIGS. 39-41 . Thecolor/hash code 2600 of FIG. 26 also applies to FIGS. 39-41 . FIGS.39-41 show progressively smaller spacings between antennas of theantenna system 400 and corresponding progressively larger pixel sizes.For systems in which a smaller field of view is manageable or desirable,keeping a larger antenna spacing is also an option. Instead of pushingthe grating lobes outwards until they disappear at the edge of anglespace by decreasing the antenna spacing as in FIGS. 38-41 , the gratinglobes can instead be suppressed while the broadband main lobes are keptand the size of the field of view maintained by using a lens array oranother antenna-wise light collecting method.

Continuous Medium Delay Lines

Although the feedline groups 2342-2344 between first and second stages2302 and 2304 have been described as being made up of discretefeedlines, in another embodiment, an exemplary portion of which is shownin FIG. 44 , an electromagnetic coupling between the first and secondstages 2302 and 2304 is provided by a continuous bulk medium 4400, suchas oil, air or a full or partial vacuum. The medium 4400 extendscontinuously vertically between the two stages 2302 and 2304, hererepresented by two Rotman lenses 4402 and 4404, and horizontally atleast between two boundaries 4406 and 4408 that extend parallel to adirection of propagation of EM signals between the two Rotman lenses4402 and 4404. All the EM signals that propagate between the two Rotmanlenses 4402 and 4404 propagate within a width defined by the twoboundaries 4406 and 4408.

One of the two Rotman lenses 4404 is disposed at a non-zero angle,relative to the other Rotman lens 4402, so the two Rotman lenses 4402and 4404 are not parallel to each other. Consequently, distancestraveled by EM signals between the two Rotman lenses 4402 and 4404 varymonotonically laterally 4409 across the medium 4400, as indicated bylengths of arrows 4410, 4412, 4414 and 4416. The arrows 4410-4416represent respective paths taken by the EM signals between the twoRotman lenses 4402 and 4404. In some embodiments, Ulexite(NaCaB₅O₆(OH)₆.5H₂O, hydrated sodium calcium borate hydroxide),sometimes known as TV rock, may be used for the medium 4400.

Other Considerations

FIG. 42 is a scale drawing of a prototype 4×4 crossover-free antennasystem 400 capable of two-dimensional beamforming, fabricated on aphotonic integrated circuit, including features described herein.

As noted with respect to FIGS. 15 and 16 , either the array of antennas402 or the pixel array is tilted. However, if the relative path delaysbetween first and second stage beamforming networks are multiplied by acorrection factor of

${1 + \left( \frac{1}{N} \right)^{2}},$both the array of antennas and the pixel array may be square.

More generally, for an N×M parallelogram-shaped antenna array 402, themaximum difference in delay (scaling factor) in the two-dimensionalbeamforming network 420 is calculated according to equation (14):

$\begin{matrix}{{{Scaling}{factor}} = {2 \times \left( \frac{N - 1}{2} \right) \times \left( \frac{M - 1}{2} \right)}} & (14)\end{matrix}$

To make it into a square device, i.e., both the antennas and the farfield pixels are rectangular, rather than parallelograms, multiply thedelay lengths by the correction factor of

$1 + {\left( \frac{1}{N} \right)^{2}.}$Thus, the scaling factor equation for an N×M, i.e., N rows by M columns,antenna array 402 becomes:

$\begin{matrix}{{{Scaling}{factor}} = {2 \times \left( \frac{N - 1}{2} \right) \times \left( \frac{M - 1}{2} \right) \times \left( {1 + \left( \frac{1}{N} \right)^{2}} \right)}} & (15)\end{matrix}$

As examples: a 4×5 rectangular antenna array 402 has a maximum delaydifference of 6.375; and a 4×4 parallelogram-shaped antenna array 402has a maximum delay difference of 4.5.

The three stages 2302-2306 (FIG. 23 ) of the two-dimensional beamformingnetwork 420 can be reversed, i.e., the Rotman lens 422 of the thirdstage 2306 can be coupled to the antenna array 402, and the Rotmanlenses 2362-2364 of the first stage 2302 can be coupled to the beamports 428-432. The two-dimensional beamforming network 420 implements acentered, discrete, two-dimensional Fourier transform, which is the sameas its transpose, i.e. inputs and outputs swapped. The individualFourier transform devices that make up the first, second, and thirdlayers have a similar property in that they also function similarlyforwards as backwards, so one or more of them may be reversed, withoutmodifying the function of the overall device.

As with Rotman lens-based systems, signals can be simultaneously appliedto more than one of the ports 428-432, or more than one segment, of thebeam-side interface 426, and the antenna systems 400, 2300, 2500 andbeamforming networks 420 described herein simultaneously producemultiple radiated beams, each in a different direction. Similarly, dueto the reciprocity theorem, multiple signals simultaneously receivedfrom respective different directions by the antenna array 402 causeoutput signals to be simultaneously available at the corresponding ports428-432, or segments, of the beam-side interface 426. In both receiveand transmit cases, the multiple simultaneous signals need notnecessarily have equal amplitudes or, especially in the case ofbroadband embodiments, equal wavelengths.

One advantage provided by embodiments of the present invention lies inthe fact that increasing the number of pixels, or beams or antennas,along each dimension of the field of view 500 (FIG. 5 ) does notnecessarily increase the number of distinct elementary components thatare traversed by a photon passing through the antenna system 400, 2300,2500. The two-dimensional beamforming network 420 can be considered asincluding a plurality of distinct elementary components. As used herein,a distinct elementary component is a signal handling component, such asa Rotman lens, a planar crossover, a phase shifter, a grating coupler, aresonator, or a waveguide. The word “elementary” is used to prevent theentire beamforming network being considered a single component. The word“distinct” is used, because some embodiments include multiple instancesof a given component. For example, the antenna system 2300 describedwith respect to FIG. 23 includes four Rotman lenses 2362-2364 in thefour first beamforming networks 2308-2312.

Each dimension of a grating lobe-free two-dimensional field of view502-508 (FIG. 5 ) includes a respective number of pixels. In someembodiments, the two-dimensional beamforming network (420) is configuredsuch that photons traverse, on average, a number of the distinctelementary components that is essentially constant, with respect to thenumber of pixels, or antennas or resolution, along each dimension of thegrating lobe-free field of view (502-508). “Constant, with respect tothe number of pixels” means that increasing the number of pixels, orbeams or antennas, along each dimension does not necessarily increasethe number of traversed components. This is valuable, since losses occurat component interfaces. Thus, increasing the number of pixels, or beamsor antennas, along each dimension does not increase the total loss fromcomponent interfaces. As used herein, the term “photon” has itsconventional meaning, i.e., the term photon applies to both light andradio frequency electromagnetic signals.

“On average” means relative to the amount of light, i.e., photons orflux. When we say a “photon passing through the antenna system,” we meanthe photon through the entire antenna system, between an antenna 404-410and a port 428-432, or segment, of the beam-side interface 426.

While the invention is described through the above-described exemplaryembodiments, modifications to, and variations of, the illustratedembodiments may be made without departing from the inventive conceptsdisclosed herein. For example, although specific parameter values, suchas dimensions and materials, may be recited in relation to disclosedembodiments, within the scope of the invention, the values of allparameters may vary over wide ranges to suit different applications.Unless otherwise indicated in context, or would be understood by one ofordinary skill in the art, terms such as “about” mean within ±20%.Terms, such as “equal path length,” “equal to intra-element columnspacing,” and “parallel” mean within manufacturing tolerances.

As used herein, including in the claims, the term “and/or,” used inconnection with a list of items, means one or more of the items in thelist, i.e., at least one of the items in the list, but not necessarilyall the items in the list. As used herein, including in the claims, theterm “or,” used in connection with a list of items, means one or more ofthe items in the list, i.e., at least one of the items in the list, butnot necessarily all the items in the list. “Or” does not mean “exclusiveor.”

Although aspects of embodiments may be described with reference toflowcharts and/or block diagrams, functions, operations, decisions, etc.of all or a portion of each block, or a combination of blocks, may becombined, separated into separate operations or performed in otherorders. References to a “module” are for convenience and not intended tolimit its implementation. All or a portion of each block, module orcombination thereof may be implemented as computer program instructions(such as software), hardware (such as combinatorial logic, ApplicationSpecific Integrated Circuits (ASICs), Field-Programmable Gate Arrays(FPGAs), processor or other hardware), firmware or combinations thereof.

The Rotman lenses, or portions thereof, may be implemented by one ormore processors executing, or controlled by, instructions stored in amemory. Each processor may be a general purpose processor, such as acentral processing unit (CPU), a graphic processing unit (GPU), digitalsignal processor (DSP), a special purpose processor, etc., asappropriate, or combination thereof.

The memory may be random access memory (RAM), read-only memory (ROM),flash memory or any other memory, or combination thereof, suitable forstoring control software or other instructions and data. Instructionsdefining the functions of the present invention may be delivered to aprocessor in many forms, including, but not limited to, informationpermanently stored on tangible non-transitory non-writable storage media(e.g., read-only memory devices within a computer, such as ROM, ordevices readable by a computer I/O attachment, such as CD-ROM or DVDdisks), information alterably stored on tangible non-transitory writablestorage media (e.g., floppy disks, removable flash memory and harddrives) or information conveyed to a computer through a communicationmedium, including wired or wireless computer networks. Moreover, whileembodiments may be described in connection with various illustrativedata structures, systems may be embodied using a variety of datastructures.

Disclosed aspects, or portions thereof, may be combined in ways notlisted above and/or not explicitly claimed. In addition, embodimentsdisclosed herein may be suitably practiced, absent any element that isnot specifically disclosed herein. Accordingly, the invention should notbe viewed as being limited to the disclosed embodiments.

As used herein, numerical terms, such as “first,” “second” and “third,”are used to distinguish respective beamforming networks from one anotherand are not intended to indicate any particular order or total number ofbeamforming networks in any particular embodiment. Thus, for example, agiven embodiment may include only a second beamforming network and athird beamforming network.

What is claimed is:
 1. An antenna system comprising: an array ofantennas disposed in a predetermined non-linear pattern, the array ofantennas comprising a plurality of antennas and having a two-dimensionalfield of view; and a two-dimensional beamforming network (BFN) disposedentirely on a single first surface, and having a one-dimensionalarray-side interface disposed on the first surface and a one-dimensionalbeam-side interface disposed on the first surface; wherein: the antennasof the array of antennas are individually communicably coupled to thearray-side interface, such that segments of the beam-side interface mapto respective pixels in the two-dimensional field of view; thepredetermined non-linear pattern defines a second surface; the array ofantennas comprises a plurality of disjoint sets of antennas; eachdisjoint set of antennas comprises a plurality of antennas of the arrayof antennas; and for each disjoint set of antennas, each antenna of atleast a non-empty subset of the disjoint set of antennas isperpendicularly displaced a respective distance along the second surfacefrom a longitudinal axis of a hypothetical linear array of antennasdisposed on the second surface.
 2. An antenna system according to claim1, wherein the array of antennas and the two-dimensional beamformingnetwork collectively form a true time delay system.
 3. An antenna systemaccording to claim 1, wherein the one-dimensional array-side interfaceis segmented.
 4. An antenna system according to claim 1, wherein theone-dimensional beam-side interface is continuous.
 5. An antenna systemaccording to claim 1, wherein the first surface is planar.
 6. An antennasystem according to claim 1, wherein the first surface is non-planar. 7.An antenna system according to claim 1, wherein the first surface isfolded.
 8. An antenna system according to claim 1, wherein thepredetermined non-linear pattern defines a second surface that extendssmoothly from an edge of the first surface.
 9. An antenna systemaccording to claim 8, wherein: the predetermined non-linear patterndefines a second surface; and the array of antennas is communicablycoupled to the array-side interface via a crossover-free networkdisposed entirely on the first and/or second surface.
 10. An antennasystem according to claim 1, wherein: the antenna system has a designwavelength λ; the plurality of disjoint sets of antennas comprises Ndisjoint sets of antennas; and within each disjoint set of the antennas:the antennas are spaced apart in a direction parallel to thelongitudinal axis of the hypothetical linear array of antennas, whereinspacing between each pair of adjacent antennas is an integral multipleof about ½λ; and the antennas are spaced apart in a directionperpendicular to the longitudinal axis of the hypothetical linear arrayof antennas.
 11. An antenna system according to claim 10, wherein: theantenna system has a design wavelength λ; and within each disjoint setof the antennas: the antennas are spaced apart by respective integralmultiples of (λ/2) in the direction parallel to the longitudinal axis;and the antennas are spaced apart by respective integral multiples of(λ/2) in the direction perpendicular to the longitudinal axis.
 12. Anantenna system according to claim 1, wherein: the predeterminednon-linear pattern defines a second surface; the array of antennascomprises a plurality of disjoint sets of antennas; each disjoint set ofantennas comprises a plurality of antennas of the array of antennas; andfor each disjoint set of antennas, each antenna of at least a non-emptysubset of the disjoint set of antennas is displaced a respectivedistance, measured along the second surface and parallel to onedimension of the two-dimensional field of view, from a longitudinal axisof a hypothetical linear array of antennas disposed on the secondsurface.
 13. An antenna system according to claim 12, wherein: theantenna system has a design wavelength λ; the plurality of disjoint setsof antennas comprises N disjoint sets of antennas; and within eachdisjoint set of the antennas: the antennas are spaced apart in adirection parallel to the longitudinal axis of the hypothetical lineararray of antennas, wherein spacing between each pair of adjacentantennas is an integral multiple of about ½λ; and the antennas arespaced apart in a direction perpendicular to the longitudinal axis ofthe hypothetical linear array of antennas.
 14. An antenna systemaccording to claim 12, wherein: the antenna system has a designwavelength λ; and within each disjoint set of the antennas: the antennasare spaced apart by respective integral multiples of (λ/2) in thedirection parallel to the longitudinal axis; and the antennas are spacedapart by respective integral multiples of (λ/2) in the directionperpendicular to the longitudinal axis.
 15. An antenna system accordingto claim 1, wherein the antenna system has a design wavelength betweenabout 10 nanometers and about 1 millimeter.
 16. An antenna systemaccording to claim 1, wherein the antenna system has a design wavelengthbetween about 1 millimeter and about 100 meters.
 17. An antenna systemaccording to claim 1, wherein: the predetermined non-linear patterndefines a second surface; and each antenna of the array of antennascomprises a grating coupler configured to optically couple to free spacebeyond the second surface with a coupling efficiency of at least about25%.
 18. An antenna system according to claim 1, wherein thetwo-dimensional beamforming network comprises a Rotman lens.
 19. Anantenna system according to claim 1, wherein the two-dimensionalbeamforming network comprises a Fourier transformer.
 20. An antennasystem according to claim 1, wherein the two-dimensional beamformingnetwork comprises a Butler matrix.
 21. An antenna system according toclaim 1, wherein the two-dimensional beamforming network comprises asingle-stage beamforming network.
 22. An antenna system according toclaim 1, wherein the two-dimensional beamforming network comprises asingle Rotman lens.
 23. An antenna system according to claim 1, wherein:the array of antennas comprises N (N>1) disjoint sets of antennas; eachdisjoint set of antennas comprises a plurality of antennas of the arrayof antennas; and the two-dimensional beamforming network comprises: Nfirst beamforming networks, each first beamforming network beingassociated with a distinct set of the antennas and having a beam-sideinterface and a plurality of array-side ports, wherein the array-sideports of each first beamforming network are individually communicablycoupled to respective antennas of the associated set of the antennas,the array-side ports of the N first beamforming networks therebycollectively forming the one-dimensional array-side interface of thetwo-dimensional beamforming network; N second beamforming networks, eachsecond beamforming network being associated with a distinct firstbeamforming network and having an array-side interface and a beam-sideinterface, wherein the beam-side interface of each second beamformingnetwork is communicably coupled to the beam-side interface of theassociated first beamforming network; and a third beamforming networkhaving an array-side interface and a plurality of beam-side ports,wherein the array-side interface of each second beamforming network iscommunicably coupled to a respective distinct portion of the array-sideinterface of the third beamforming network, and the plurality ofbeam-side ports of the third beamforming network collectively forms theone-dimensional beam-side interface of the two-dimensional beamformingnetwork.
 24. An antenna system according to claim 23, wherein: eachfirst beamforming network consists essentially of a respectiveone-dimensional beamforming network; each second beamforming networkconsists essentially of a respective one-dimensional beamformingnetwork; and the third beamforming network consists essentially of adistinct one-dimensional beamforming network.
 25. An antenna systemaccording to claim 23, wherein: each first beamforming network consistsessentially of a respective Rotman lens; each second beamforming networkconsists essentially of a respective Rotman lens; and the thirdbeamforming network consists essentially of a distinct Rotman lens. 26.An antenna system according to claim 23 wherein, for each firstbeamforming network: the array-side ports of the first beamformingnetwork are transversely ordered in a first order; the antennas of theassociated set of the antennas are transversely ordered in a secondorder; and the antennas of the associated set of the antennas areindividually communicably coupled to the respective array-side portssuch that the first order is opposite the second order.
 27. An antennasystem according to claim 23, wherein: the antenna system has a designwavelength λ; and the beam-side interface of each second beamformingnetwork is communicably coupled to the beam-side interface of theassociated first beamforming network by a respective associated firstcoupling, thereby collectively defining a plurality of first couplings,wherein each non-central first coupling is configured to delay signalsof wavelength λ, propagating therethrough by a respective relative delayamount, such that the delay amount varies monotonically transverselyacross the non-central first coupling.
 28. An antenna system accordingto claim 27, wherein: each first beamforming network is numbered with aunique integer j between $- \left\lfloor \frac{N}{2} \right\rfloor$ and${+ \left\lfloor \frac{N}{2} \right\rfloor};$ and for each non-centralfirst coupling, the delay amount varies monotonically transverselyacross the non-central first coupling between about${+ \left( \frac{N - 1}{2} \right)}j\lambda$ and about${- \left( \frac{N - 1}{2} \right)}j\lambda$
 29. An antenna systemaccording to claim 28 wherein, for each central first coupling, therelative delay amount is about zero.
 30. An antenna system according toclaim 28, wherein each first coupling comprises a respective pluralityof discrete waveguides.
 31. An antenna system according to claim 28wherein, for each first beamforming network: the set of the antennasassociated with the first beamforming network comprises a respectivenumber M of antennas; and the associated first coupling comprises Mdiscrete waveguides.
 32. An antenna system according to claim 28,wherein each non-central first coupling comprises a respective mediumconfigured to delay signals propagating therethrough by a respectivedelay amount, such that the delay amount varies continuouslytransversely across the medium.
 33. An antenna system according to claim1, wherein: each dimension of a grating lobe-free two-dimensional fieldof view comprises a respective number of pixels; the two-dimensionalbeamforming network comprises a plurality of distinct elementarycomponents; and the two-dimensional beamforming network is configuredsuch that photons traverse, on average, a number of the distinctelementary components that is constant, with respect to the number ofpixels along each dimension of the grating lobe-free field of view.